angles and triangles homework 1

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Unit 2: Angles

About this unit.

In this topic, we will learn what an angle is and how to label, measure and construct them. We will also explore special types of angles.

Angle introduction

• Angles: introduction (Opens a modal)
• Naming angles (Opens a modal)
• Angle basics review (Opens a modal)
• Angle basics 4 questions Practice
• Name angles 4 questions Practice

Measuring angles

• Measuring angles in degrees (Opens a modal)
• Measuring angles using a protractor (Opens a modal)
• Measuring angles using a protractor 2 (Opens a modal)
• Measuring angles review (Opens a modal)
• Measure angles 4 questions Practice

Constructing angles

• Constructing angles (Opens a modal)
• Constructing angles review (Opens a modal)
• Draw angles 7 questions Practice

Angles in circles

• Angle measurement & circle arcs (Opens a modal)
• Angles in circles word problems (Opens a modal)
• Angles in circles 7 questions Practice

Angle types

• Recognizing angles (Opens a modal)
• Drawing acute, right and obtuse angles (Opens a modal)
• Identifying an angle (Opens a modal)
• Angle types review (Opens a modal)
• Angle types 4 questions Practice
• Recognize angles in figures 4 questions Practice
• Draw right, acute, and obtuse angles 7 questions Practice
• Benchmark angles 7 questions Practice

Vertical, complementary, and supplementary angles

• Complementary & supplementary angles (Opens a modal)
• Complementary and supplementary angles review (Opens a modal)
• Vertical angles review (Opens a modal)
• Angle relationships example (Opens a modal)
• Vertical angles are congruent proof (Opens a modal)
• Identifying supplementary, complementary, and vertical angles 7 questions Practice
• Complementary and supplementary angles (visual) 4 questions Practice
• Complementary and supplementary angles (no visual) 7 questions Practice
• Vertical angles 4 questions Practice

Angles between intersecting lines

• Angles, parallel lines, & transversals (Opens a modal)
• Parallel & perpendicular lines (Opens a modal)
• Missing angles with a transversal (Opens a modal)
• Parallel lines & corresponding angles proof (Opens a modal)
• Missing angles (CA geometry) (Opens a modal)
• Proving angles are congruent (Opens a modal)
• Proofs with transformations (Opens a modal)
• Angle relationships with parallel lines 7 questions Practice
• Line and angle proofs 4 questions Practice

Sal's old angle videos

• Intro to angles (old) (Opens a modal)
• Angles (part 2) (Opens a modal)
• Angles (part 3) (Opens a modal)
• Angles formed between transversals and parallel lines (Opens a modal)
• Angles of parallel lines 2 (Opens a modal)
• The angle game (Opens a modal)
• The angle game (part 2) (Opens a modal)
• Acute, right, & obtuse angles (Opens a modal)

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Complementary, Supplementary, & Vertical Angles

The geometry worksheets on this page can be used to introduce and review the concepts of complementary, supplementary angles, and vertical angles.

Most worksheets on this page align with Common Core Standard 7.G.B.5.

Complementary & Supplementary Angles

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Complementary Angles

Supplementary angles, adding adjacent angles together.

Here you'll find a collection of printables for teaching students how to measure angles with a protractor.

This page has many worksheets on calculating the areas of triangles.

Here you'll find a selection of worksheets for teaching surface area.

Here's our geometry index page that will link you to all geometry topics on our site.

Sample Worksheet Images

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Chapter 2: Trigonometric Ratios

Exercises: 2.3 Solving Right Triangles

Suggested Homework Problems

Exercises Homework 2.3

Exercise group.

For Problems 1–4, solve the triangle. Round answers to hundredths.

For Problems 5–10,

• Sketch the right triangle described.
• Solve the triangle.

[latex]A = 42{^o}, c = 26[/latex]

[latex]B = 28{^o}, c = 6.8[/latex]

[latex]B = 33{^o}, a = 300[/latex]

[latex]B = 79{^o}, a = 116[/latex]

[latex]A = 12{^o}, a = 4[/latex]

[latex]A = 50{^o}, a = 10[/latex]

For Problems 11–16,

• Without doing the calculations, list the steps you would use to solve the triangle.

[latex]B = 53.7{^o}, b = 8.2[/latex]

[latex]B = 80{^o}, a = 250[/latex]

[latex]A = 25{^o}, b = 40[/latex]

[latex]A = 15{^o}, c = 62[/latex]

[latex]A = 64.5{^o}, c = 24[/latex]

[latex]B = 44{^o}, b = 0.6[/latex]

For Problems 17–22, find the labeled angle. Round your answer to tenths of a degree.

For Problems 23–28, evaluate the expression and sketch a right triangle to illustrate.

[latex]\sin^{-1} 0.2[/latex]

[latex]\cos^{-1} 0.8[/latex]

[latex]\tan^{-1} 1.5[/latex]

[latex]\tan^{-1} 2.5[/latex]

[latex]\cos^{-1} 0.2839[/latex]

[latex]\sin^{-1} 0.4127[/latex]

For Problems 29–32, write two different equations for the statement.

The cosine of [latex]15 {^o}[/latex] is [latex]0.9659\text{.}[/latex]

The sine of [latex]70 {^o}[/latex] is [latex]0.9397\text{.}[/latex]

The angle whose tangent is [latex]3.1445[/latex] is [latex]65 {^o}\text{.}[/latex]

The angle whose cosine is [latex]0.0872[/latex] is [latex]85 {^o}\text{.}[/latex]

Evaluate the expressions and explain what each means. [latex]\begin{equation*} \sin^{-1} (0.6), (\sin 6{^o})^{-1} \end{equation*}[/latex]

Evaluate the expressions and explain what each means. [latex]\begin{equation*} \cos^{-1} (0.36), (\cos 36{^o})^{-1} \end{equation*}[/latex]

For Problems 35–38,

• Sketch a right triangle that illustrates the situation. Label your sketch with the given information.
• Choose the appropriate trig ratio and write an equation, then solve the problem.

The gondola cable for the ski lift at Snowy Peak is [latex]2458[/latex] yards long and climbs [latex]1860[/latex] feet. What angle with the horizontal does the cable make?

The Leaning Tower of Pisa is [latex]55[/latex] meters in length. An object dropped from the top of the tower lands [latex]4.8[/latex] meters from the base of the tower. At what angle from the horizontal does the tower lean?

A mining company locates a vein of minerals at a depth of [latex]32[/latex] meters. However, there is a layer of granite directly above the minerals, so they decide to drill at an angle, starting [latex]10[/latex] meters from their original location. At what angle from the horizontal should they drill?

The birdhouse in Carolyn’s front yard is [latex]12[/latex] feet tall, and its shadow at [latex]4[/latex] pm is [latex]15[/latex] feet [latex]4[/latex] inches long. What is the angle of elevation of the sun at [latex]4[/latex] pm?

For Problems 39–42,

[latex]a = 18, b = 26[/latex]

[latex]a = 35, b = 27[/latex]

[latex]b = 10.6 , c = 19.2[/latex]

[latex]a = 88, c = 132[/latex]

For Problems 43–48,

• Make a sketch that illustrates the situation. Label your sketch with the given information.
• Write an equation and solve the problem.

The Mayan pyramid of El Castillo at Chichén Itzá in Mexico has [latex]91[/latex] steps. Each step is 26 cm high and 30 cm deep.

• What angle does the side of the pyramid make with the horizontal?
• What is the distance up the face of the pyramid, from base to top platform?

An airplane begins its descent when its altitude is 10 kilometers. The angle of descent should be [latex]3{^o}[/latex] from horizontal.

• How far from the airport (measured along the ground) should the airplane begin its descent?
• How far will the airplane travel on its descent to the airport?

A communications satellite is in a low earth orbit (LOE) at an altitude of 400 km. From the satellite, the angle of depression to earth’s horizon is [latex]19.728{^o}\text{.}[/latex] Use this information to calculate the radius of the earth.

The first Ferris wheel was built for the [latex]1893[/latex] Chicago World’s Fair. It had a diameter of [latex]250[/latex] feet, and the boarding platform, at the base of the wheel, was [latex]14[/latex] feet above the ground. If you boarded the wheel and rotated through an angle of [latex]50{^o}\text{,}[/latex] what would be your height above the ground?

To find the distance across a ravine, Delbert takes some measurements from a small airplane. When he is a short distance from the ravine at an altitude of [latex]500[/latex] feet, he finds that the angle of depression to the near side of the ravine is [latex]56{^o}\text{,}[/latex] and the angle of depression to the far side is [latex]32{^o}\text{.}[/latex] What is the width of the ravine? (Hint: First find the horizontal distance from Delbert to the near side of the ravine.)

The window in Francine’s office is [latex]4[/latex] feet wide and [latex]5[/latex] feet tall. The bottom of the window is 3 feet from the floor. When the sun is at an angle of elevation of [latex]64{^o}\text{,}[/latex] what is the area of the sunny spot on the floor?

Which of the following numbers are equal to [latex]\cos 45{^o}\text{?}[/latex]

• [latex]\displaystyle \dfrac{\sqrt{2}}{2}[/latex]
• [latex]\displaystyle \dfrac{1}{\sqrt{2}}[/latex]
• [latex]\displaystyle \dfrac{2}{\sqrt{2}}[/latex]
• [latex]\displaystyle \sqrt{2}[/latex]

Which of the following numbers are equal to [latex]\tan 30{^o}\text{?}[/latex]

• [latex]\displaystyle \sqrt{3}[/latex]
• [latex]\displaystyle \dfrac{1}{\sqrt{3}}[/latex]
• [latex]\displaystyle \dfrac{\sqrt{3}}{3}[/latex]
• [latex]\displaystyle \dfrac{3}{\sqrt{3}}[/latex]

Which of the following numbers are equal to [latex]\tan 60{^o}\text{?}[/latex]

Which of the following numbers are equal to [latex]\sin 60{^o}\text{?}[/latex]

• [latex]\displaystyle \dfrac{3}{\sqrt{2}}[/latex]
• [latex]\displaystyle \dfrac{\sqrt{3}}{2}[/latex]
• [latex]\displaystyle \dfrac{\sqrt{2}}{3}[/latex]
• [latex]\displaystyle \dfrac{2}{\sqrt{3}}[/latex]

For Problems 53–58, choose all values from the list below that are exactly equal to, or decimal approximations for, the given trig ratio. (Try not to use a calculator!)

[latex]\cos 30{^o}[/latex]

[latex]\sin 45{^o}[/latex]

[latex]\tan 30{^o}[/latex]

[latex]\cos 60{^o}[/latex]

[latex]\sin 90{^o}[/latex]

[latex]\cos 0{^o}[/latex]

Fill in the table from memory with exact values. Do you notice any patterns that might help you memorize the values?

Fill in the table from memory with decimal approximations to four places.

For Problems 61 and 62, compare the given value with the trig ratios of the special angles to answer the questions. Try not to use a calculator.

Is the acute angle larger or smaller than [latex]45{^o}\text{?}[/latex]

• [latex]\displaystyle \sin \alpha = 0.7[/latex]
• [latex]\displaystyle \tan \beta = 1.2[/latex]
• [latex]\displaystyle \cos \gamma = 0.65[/latex]

Is the acute angle larger or smaller than [latex]60{^o}\text{?}[/latex]

• [latex]\displaystyle \cos \theta = 0.75[/latex]
• [latex]\displaystyle \tan \phi = 1.5[/latex]
• [latex]\displaystyle \sin \psi = 0.72[/latex]

For Problems 63–72, solve the triangle. Give your answers as exact values.

• Find the perimeter of a regular hexagon if the apothegm is [latex]8[/latex] cm long. (The apothegm is the segment from the center of the hexagon and perpendicular to one of its sides.)
• Find the area of the hexagon.

Triangle [latex]ABC[/latex] is equilateral, and its angle bisectors meet at point [latex]P\text{.}[/latex] The sides of [latex]\triangle ABC[/latex] are 6 inches long. Find the length of [latex]AP\text{.}[/latex]

Find an exact value for the area of each triangle.

Find an exact value for the perimeter of each parallelogram.

• Find the area of the outer square.
• Find the dimensions and the area of the inner square.
• What is the ratio of the area of the outer square to the area of the inner square?
• Find the area of the inner square.
• Find the dimensions and the area of the outer square.

Trigonometry Copyright © 2024 by Bimal Kunwor; Donna Densmore; Jared Eusea; and Yi Zhen. All Rights Reserved.

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Big Ideas Math Answers Grade 8 Chapter 3 Angles and Triangles

The students of middle school can get the Solution Key for Big Ideas Math Grade 8 Chapter 3 Angles and Triangles on this page. With the help of this Big Ideas Math Book 8th Grade Answer Key Chapter 3 Angles and Triangles you can finish your homework in time and also improve your performance in the exams. Get free step by step solutions for all the questions in Big Ideas Math Answers Grade 8 Chapter 3 Angles and Triangles.

Big Ideas Math Book 8th Grade Answer Key Chapter 3 Angles and Triangles

Download Big Ideas Math Answers Grade 8 Chapter 3 Angles and Triangles pdf for free of cost. The solutions for each and every question is prepared in an easy manner. Go through the table of contents shown in the below section to know the topics covered in Big Ideas Math Answers Grade 8 Chapter 3 Angles and Triangles.

Performance

Angles and Triangles STEAM Video/Performance

Angles and triangles getting ready for chapter 3.

Lesson: 1 Parallel Lines and Transversals

Lesson 3.1 Parallel Lines and Transversals

Parallel lines and transversals homework & practice 3.1.

Lesson: 2 Angles of Triangles

Lesson 3.2 Angles of Triangles

Angles of triangles homework & practice 3.2.

Lesson: 3 Angles of Polygons

Lesson 3.3 Angles of Polygons

Angles of polygons homework & practice 3.3.

Lesson: 4 Using Similar Triangles

Lesson 3.4 Using Similar Triangles

Using similar triangles homework & practice 3.4.

Chapter 3 – Angles and Triangles

Angles and Triangles Connecting Concepts

Angles and triangles chapter review, angles and triangles practice test, angles and triangles cumulative practice.

STEAM Video

Watch the STEAM Video “Honeycombs.” Then answer the following questions.

Answer: The sum of interior angles of the equilateral triangle = 180° x + x + x = 180° 3x° = 180° x = 180/3 x° = 60°

Performance Task

Turtle Shells

Chapter Exploration

Answer: 8 angles are formed by the parallel lines and the transversal b. Which of these angles have equal measures? Explain your reasoning.

EXPLORATION 1

Use the figure to find the measure of the angle. Explain your reasoning

Question 1. ∠1

Answer: 63°

Question 2. ∠2

Answer: 117°

Answer: ∠1 and 59° are the supplementary angles ∠1 + 59° = 180° ∠1 = 180° – 59° ∠1 = 121° ∠2 and 59° are vertical angles. They are congruent. So, the measure of ∠1 is 121° ∠3 and 59° are supplementary angles. ∠3 + 59° = 180° ∠3 = 180° – 59° ∠3 = 121° ∠4, ∠5, ∠6, ∠7 corresponding angles are congruent because they are formed by a transversal intersecting parallel side. the measure of  ∠4 is 121° the measure of ∠5 is 59° the measure of  ∠6 is 121° the measure of ∠7 is 59°

In Example 3, the measure of ∠4 is 84°. Find the measure of the angle. Explain your reasoning.

Question 4. ∠3

Question 5. ∠5

Answer: ∠4 and ∠5 are alternate interior angles formed by transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠5 is 84°

Question 6. ∠6

Answer: ∠3 and ∠6 are alternate exterior angles formed by transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠6 is 96°

Self-Assessment for Concepts & Skills Solve each exercise. Then rate your understanding of the success criteria in your journal.

FINDING ANGLE MEASURES Use the figure to find the measures of the numbered angles.

Answer: ∠1 and 120° are the supplementary angles. ∠1 + 120° = 180° ∠1 = 180 – 120 ∠1 = 60° Thus the measure of ∠1 is 60° ∠2 and 120° are the vertical angles. They are congruent. Thus the measure of ∠2 is 120° ∠3 and 120° are the supplementary angles. ∠3 + 120° = 180° ∠3 = 180 – 120 ∠3 = 60° ∠4, ∠5, ∠6, ∠7 are corresponding angles are formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠4 is 60° Thus the measure of ∠5 is 120° Thus the measure of ∠6 is 120° Thus the measure of ∠7 is 60°

Answer: ∠1 and 35° are the supplementary angles. ∠1 + 35° = 180° ∠1 = 180 – 35 ∠1 = 145° Thus the measure of ∠1 is 145° ∠2 and 35° are the supplementary angles. ∠2 + 35° = 180° ∠2 = 180 – 35 ∠2 = 145° Thus the measure of ∠2 is 145° ∠3 and 35° are the vertical angles. They are congruent. Thus the measure of ∠3 is 35° ∠4, ∠5, ∠6, ∠7 are corresponding angles are formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠4 is 35° Thus the measure of ∠5 is 145° Thus the measure of ∠6 is 145° Thus the measure of ∠7 is 35°

Answer: ∠2, ∠6 are corresponding angles are formed by transversal intersecting parallel lines. ∠6, ∠8 are vertical angles are formed by transversal intersecting parallel lines. ∠5 does not belong to the other three because all the other three measure are equal.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Answer: The angle a and the angle of 70 degrees are complementary angles because they belong to a right triangle, where the third angle is the right angle. ∠a + 70 = 90 ∠a = 90 – 70 ∠a = 20°

Answer: The lines AB and CD are parallel. ABC and BCD are the corresponding angles formed by transversal intersecting parallel lines. ∠BCD = 55° ∠BAC + ∠ABC + ∠ACB = 180° The sum of the angles in a triangle is 180° ∠BAC + 55°+ 52° = 180° ∠BAC + 107° = 180° ∠BAC = 180° – 107° ∠BAC = 73° So, the head tube angle of a bike is 73°

Review & Refresh

Find the values of the ratios (red to blue) of the perimeters and areas of the similar figures.

Answer: perimeter of red hexagon/perimeter of blue hexagon = \(\frac{3}{5}\) The values of the ratios of the perimeter is \(\frac{3}{5}\) Area of red hexagon/Area of blue hexagon = (\(\frac{3}{5}\))² = \(\frac{9}{25}\) The values of the ratios of the area is \(\frac{9}{25}\)

Answer: perimeter of red trapezium /perimeter of blue trapezium = \(\frac{7}{6}\) The values of the ratios of the perimeter is \(\frac{7}{6}\) Area of red hexagon/Area of blue hexagon = (\(\frac{7}{6}\))² = \(\frac{49}{36}\) The values of the ratios of the area is \(\frac{49}{36}\)

Evaluate the expression.

Question 3. 4 + 3 2

Answer: 4 + 9 = 13

Question 4. 5(2) 2 – 6

Answer: 5(4) – 6 20 – 6 = 14

Question 5. 11 + (-7) 2 – 9

Answer: 11 + 49 – 9 11 + 40 = 50

Concepts, Skills, & Problem Solving EXPLORING INTERSECTIONS OF LINES Use a protractor to determine whether lines a and b are parallel. ( See Exploration 1, p. 103.)

Answer: Use a protractor to measure ∠1 and ∠2 ∠1 ≈ 60° ∠2 ≈ 60° ∠1 and ∠2, it means the two angles are congruent. The angles are exterior alternate angles. According to the converse of the exterior alternate angles theorem, the two lines are parallel. a || b

Answer: Use a protractor to measure ∠1 and ∠2 ∠1 ≈ 50° ∠2 ≈ 60° ∠1 and ∠2, it means the two angles are not congruent. The angles are exterior alternate angles. According to the converse of the exterior alternate angles theorem, the two lines are not parallel.

FINDING ANGLE MEASURES Use the figure to find the measures of the numbered angles. Explain your reasoning.

Answer: ∠1 and 107° are corresponding angles. They are congruent. So, the measure of ∠1 is 107°. ∠1 and ∠2 are supplementary angles. ∠1 + ∠2 = 180° 107° + ∠2 = 180° ∠2 = 180° – 107° ∠2 = 73° So, the measure of ∠2 is 73°

Answer: ∠3 and 95° are corresponding angles. They are congruent. Thus the measure of ∠3 is 95° ∠3 and ∠4 are supplementary angles. ∠3 + ∠4 = 180° 95° + ∠4 = 180° ∠4 = 180 – 95 ∠4 = 85° So the measure of ∠4 is 85°

Answer: ∠5 and 49° are corresponding angles. They are congruent. So, the measure of ∠5 is 49° ∠5 and ∠6 are supplementary angles. ∠5 + ∠6 = 180° 49° + ∠6 = 180° ∠6 = 180° – 49° ∠6 = 131° So, the measure of ∠6 is 131°

Answer: Since the two lines are not parallel. Hence ∠5 is not congruent to ∠6. By this, we can say that your friend is not correct.

Answer: ∠1 and ∠2 are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. The measure of ∠1 is 60° so the measure of ∠2 is 60°

Question 13. OPEN-ENDED Describe two real-life situations that use parallel lines.

Answer: Example 1: The railroad tracks and the tram tracks are parallel lines. Example 2:  The shelves of a bookcase.

USING CORRESPONDING ANGLES Use the figure to find the measures of the numbered angles.

Answer: ∠1 and 60° are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. ∠1 and ∠2 are supplementary angles. ∠1 + ∠2 = 180° 60° + ∠2 = 180° ∠2 = 180° – 60° ∠2 = 119° So, the measure of ∠2 is 119° ∠3 and ∠1 are vertical angles. They are congruent. So, the measure of ∠3 is 61° ∠4 and ∠2 are vertical angles. They are congruent. ∠5, ∠6, ∠7 corresponding angles are congruent because they are formed by a transversal intersecting parallel lines. So, the measure of ∠5 is 119° So, the measure of ∠6 is 61° So, the measure of ∠7 is 119°

Answer: ∠1 and 99° are supplementary angles. ∠1 + 99° = 180° ∠1 = 180° – 99° ∠1 = 81° Thus the measure of ∠1 is 81° ∠2 and 99° are vertical angles. They are congruent. The measure of ∠2 is 99° ∠3 and ∠1 are vertical angles. They are congruent. So, the measure of ∠3 is 81° ∠4, ∠5, ∠6, ∠7 corresponding angles are congruent because they are formed by a transversal intersecting parallel lines. So, the measure of ∠4 is 99° So, the measure of ∠5 is 81° So, the measure of ∠6 is 99° So, the measure of ∠7 is 81°

Answer: ∠1 and 90° are supplementary angles. ∠1 + 90° = 180° ∠1 = 180° – 90° ∠1 = 90° Thus the measure of ∠1 is 90° ∠2 and 90° are vertical angles. They are congruent. Thus the measure of ∠2 is 90° ∠3 and ∠1 are vertical angles. They are congruent. So, the measure of ∠3 is 90° ∠4, ∠5, ∠6, ∠7 corresponding angles are congruent because they are formed by a transversal intersecting parallel lines. So, the measure of ∠4 is 90° So, the measure of ∠5 is 90° So, the measure of ∠6 is 90° So, the measure of ∠7 is 90°

USING CORRESPONDING ANGLES Complete the statement. Explain your reasoning.

Answer: ∠1 and ∠8 are corresponding angles. They are congruent. The measure of ∠1 = 124°, then the measure of ∠8 is 124° ∠8 and ∠4 are supplementary angles. ∠8 + ∠4 = 180° 124° + ∠4 = 180° ∠4 = 180° – 124° ∠4 = 56° So, the measure of ∠4 is 56°

Answer: ∠2 and ∠7 are corresponding angles. They are congruent. The measure of ∠2 = 48°, then the measure of ∠7 is 48° ∠7 and ∠3 are supplementary angles. ∠7 + ∠3 = 180° 48° + ∠3 = 180° ∠3 = 180° – 48° ∠3 = 132° Thus the measure of ∠3 = 132°

Answer: ∠4 and ∠2 are alternate interior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠2 is 55°

Answer: ∠6 and ∠8 are alternate exterior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠8 is 120°

Answer: ∠7 and ∠2 are corresponding angles. They are congruent. The measure of ∠7 is 50.5°, so the measure of ∠2 is 50.5° ∠2 and ∠6 are supplementary angle. ∠2 + ∠6 = 180° 50.5° + ∠6 = 180° ∠6 = 180° – 50.5° ∠6 = 129.5° So, the measure of ∠6 is 129.5°

Answer: ∠3 and ∠6 are corresponding angles. They are congruent. The measure of ∠3 is 118.7° So, the measure of ∠6 is 118.7° ∠6 and ∠2 are supplementary angles. ∠6 + ∠2 = 180° 118.7° + ∠2 = 180° ∠2 = 180° – 118.7° ∠2 = 61.3° So, the measure of ∠2 is 61.3°

Answer: ∠4 and ∠5 are alternate interior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠1 is 40°

Question 24. REASONING Is there a relationship between exterior angles that lie on the same side of a transversal? interior angles that lie on the same side of a transversal? Explain.

Question 25. REASONING When a transversal is perpendicular to two parallel lines, all the angles formed measure 90°. Explain why.

Question 26. REASONING Two horizontal lines are cut by a transversal. What is the least number of angle measures you need to know to find the measure of every angle? Explain your reasoning.

Answer: ∠1 and ∠7 are alternate exterior angles formed by a transeversal intersecting parallel lines. So, ∠1 and ∠7 are congruent. ∠1 and ∠5 are corresponding angles formed by a transeversal intersecting parallel lines. So, ∠1 and ∠5 are congruent. ∠5 and ∠7 are vertical angles so they are congruent. Hence ∠1 and ∠7 are congruent.

FINDING A VALUE Find the value of x.

Answer: ∠1 and 50° are alternate interior angles. They are congruent. So, the measure of ∠1 is 50° ∠2 and ∠1 are corresponding angles. They are congruent. So, the measure of ∠2 is 50° ∠2 and x are supplementary angle. ∠2 + x = 180° 50° + x = 180° x = 180° – 50° x = 130° So, the measure of x is 130°

Answer: ∠1 and 115° are corresponding angles. They are congruent. So, the measure of ∠1 is 115° ∠1 and x are alternate exterior angles. They are congruent. So, the measure of x is 115°

Answer: 180° rotation, translation about line t.

Question 31. OPEN-ENDED Refer to the figure. a. Do the horizontal lines appear to be parallel? Explain.

Answer: The three horizontal lines seem to spread apart, even though in reality they are parallel.

Answer: As the lines AB and CD are parallel and ∠BCD are alternate interior angles transversal BC, they are congruent. ∠ABC ≅ ∠BCD x = 64 b. How does the angle the puck hits the edge of the table relate to the angle it leaves the edge of the table?

Answer: m∠MBA + m∠ABC + m∠CBN = 180° 58 ° + 64° + m∠CBN = 180° 122 ° + m∠CBN = 180° m∠CBN = 180° – 122 ° m∠CBN = 58°

Find the measures of the interior angles of the triangle.

Answer: 81°, 25°, 74°

Explanation: Sum of all the angles in a triangle = 180° x° + 81° + 25° = 180° x° = 180° – 81° – 25° x = 74° Thus the measure of the interior angle is 74°

Answer: 43°, 51°, 86°

Explanation: Sum of all the angles in a triangle = 180° x° + (x – 35)° + 43° = 180° x° + x° – 35° + 43° = 180° 2x° + 8° = 180° 2x° = 180° – 8° 2x° = 172° x° = 172°/2 x° = 86° The measure of the interior angle of the triangle (x – 35)° = 86 – 35 (x – 35)° = 51° x° = 51° + 35° x° = 86°

Self-Assessment for Concepts & Skills

Question 4. VOCABULARY How many exterior angles does a triangle have at each vertex? Explain.

FINDING ANGLE MEASURES Find the value of x.

Answer: Sum of all the angles in a triangle = 180° x° + 25° + 40° = 180° x° + 65° = 180° x° = 180° x° = 180° – 65° x° = 115° Thus the value of x is 115°

Answer: x° = 50° + 55° x° = 105° Thus the value of x is 105°

Question 7. The Historic Triangle in Virginia connects Jamestown, Williamsburg, and Yorktown. The interior angle at Williamsburg is 120°. The interior angle at Jamestown is twice the measure of the interior angle at Yorktown. Find the measures of the interior angles at Jamestown and Yorktown. Explain your reasoning.

Answer: Given, A helicopter travels from point C to point A to perform a medical supply drop. The helicopter then needs to land at point B. A = 90° + 32° A = 122° Thus the helicopter should turn 122° at point A to travel towards point B.

Use the figure to find the measure of the angle. Explain your reasoning.

Question 1. ∠2

Answer: 82°

∠2 and 82° are alternate exterior angles formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠2 is 82°

Question 2. ∠6

∠6 and 82° are vertical angles formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠6 is 82°

Question 3. ∠4

∠4 and 82° are corresponding angles formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠4 is 82°

Question 4. ∠1

Answer: 98°

∠4 and 82° are corresponding angles formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠4 is 82° ∠4 and ∠1 are supplementary angles ∠4 + ∠1 = 180° 82° + ∠1 = 180° ∠1 = 180° – 82° ∠1 = 98°

You spin the spinner shown.

Question 5. What are the favorable outcomes of spinning a number less than 4?

Answer: 1, 2, 3

Explanation: The favorable outcome of spinning a number less than 4 is 1, 2, and 3.

Question 6. In how many ways can spinning an odd number occur?

Answer: two ways Odd numbers = 1 and 3 So, in two ways spinning an odd number can occur.

Concepts, Skills, & Problem Solving

USING PARALLEL LINES AND TRANSVERSALS Consider the figure below. (See Exploration 2, p. 111.)

Question 7. Use a protractor to find the measures of the labeled angles.

Answer: Use a protractor to determine the measures of the angles A, B, C. m∠A = 30° m∠B = 105° m∠C = 45° m∠D = 150° m∠E = 75° m∠F = 105° m∠G = 30°

Question 8. Is ∠F an exterior angle of Triangle ABC ? Justify your answer.

Answer: An exterior angle is the angle between one side of a triangle and the extension of an adjacent side. ∠F is not an exterior angle of triangle ABC because it has a side of triangle ABC, but not the extension of the adjacent side DF.

USING INTERIOR ANGLE MEASURES Find the measures of the interior angles of the triangle.

Answer: Sum of all the angles in a triangle = 180° x° + 90° + 30° = 180° x° + 120° = 180° x° = 180° – 120° x° = 60°

Answer: Sum of all the angles in a triangle = 180° x° + 65° + 40° = 180° x° + 105° = 180° x° = 180° – 105° x° = 75°

Answer: Sum of all the angles in a triangle = 180° x° + 35° + 45° = 180° x° + 80° = 180° x° = 180° – 80° x° = 100°

Answer: Sum of all the angles in a triangle = 180° x° + (x + 65)° + 25° = 180° x° + x° + 65° + 25° = 180° 2x° + 90° = 180° 2x° = 180° – 90° 2x° = 90° x° = 90°/2 x° = 45° (x + 65)° = 45 + 65 = 110 x° = 25°

Answer: Sum of all the angles in a triangle = 180° x° + (x – 44)° + 48° = 180° x° + x° – 44° + 48° = 180° 2x° + 4° = 180° 2x° = 180° – 4° 2x° = 176° x° = 176°/2 x° = 88° (x – 44)° = 88 – 44 = 44 x° = 44°

Answer: Sum of all the angles in a triangle = 180° x° + (x – 11)° + 73° = 180° x° + x° – 11° + 73° = 180° 2x° + 62° = 180° 2x° = 180° – 62° 2x° = 118° x° = 118°/2 x° = 59° (x – 11)° = 59 – 11 = 48 x° = 48°

FINDING EXTERIOR ANGLE MEASURES Find the measure of the exterior angle.

Answer: x° = 38° + 90° x° = 128° The measure of exterior angle is 128°

Answer: k° = 64° + 76° k° = 140° The measure of an exterior angle is 140°

Answer: 2a° = (a + 10°) + 44° 2a° = a + 54° 2a° – a° = 54 a° = 54 The measure of the exterior angle = 2a = 2(54°) = 108°

Answer: Sum of all the angles in a triangle = 180° x° + 75° + 75° = 180° x° + 150° – 150° = 180° – 150° x° = 30° Thus the angle that tornado direction needs to change is 30°.

Answer: Your friend is not correct because the measure of the exterior angle is equal to the sum of two non-adjacent interior angles.

Question 20. REASONING The ratio of the interior angle measures of a triangle is 2 : 3 : 5. What are the angle measures?

Answer: Sum of all the angles in a triangle = 180° 2x° + 3x° + 5x° = 180° 10x° = 180° x = 180/10 x = 18° 2x° = 2(18°) = 36° 3x° = 3(18) = 54° 5x° = 5(18) = 90°

Question 21. PROBLEM SOLVING The support for a window air-conditioning unit forms a triangle and an exterior angle. What is the measure of the exterior angle?

Answer: The measure of the exterior angle DBC is m∠DBC = m∠ABC + m∠ACB m∠ABC + m∠ACB = 90° 5x – 6 + 3x = 90 8x – 6 = 90 8x = 90 + 6 8x = 96 x = 96/8 x = 12 m∠DBC = m∠BAC+ m∠ACB = 90° + 3(12)° = 126°

Question 22. REASONING A triangle has an exterior angle with a measure of 120°. Can you determine the measures of the interior angles? Explain.

Answer: A triangle has an exterior angle with a measure of 120° m∠ACB = m∠A + m∠B m∠A + m∠B = 120° According to the exterior angles We have m∠C + m∠ACD = 180° m∠C + 120° = 180° m∠C = 180° – 120° m∠C = 60°

ANGLES OF TRIANGLES

Determine whether the statement is always, sometimes, or never true. Explain your reasoning.

Question 23. Given three angle measures, you can construct a triangle.

Answer: We can construct a triangle if the sum of the measure of the 3 angles is 180°. As a matter of fact, if the sum of the measures of the 3 angles is 180° We can build an infinity of triangles that are similar.

Question 24. The acute interior angles of a right triangle are complementary.

Answer: Let A, B, C be the angles of a right triangle with m∠A = 90° m∠A + m∠B + m∠C = 180° 90° + m∠B + m∠C = 180° m∠B + m∠C = 180° – 90° m∠B + m∠C = 90° This means ∠B and ∠C are complementary.

Question 25. A triangle has more than one vertex with an acute exterior angle.

Answer: An exterior angle of a triangle and the adjacent triangle’s angle are complementary. If an exterior angle is acute, it means the adjacent triangle’s angle is obtuse. Since we are given that more than one exterior angle is acute, it means the triangle would have more than one obtuse angle, which is impossible. The statement is never true.

Answer: The angles z and w are supplementary z + w = 180° The sum of a triangle is 180° x + y + w = 180° z = 180° – w x + y = 180° – w z = x + y

Find the sum of the interior angle measures of the green polygon.

Answer: S = (n – 2) . 180° S = (7 – 2) . 180° S = 5 . 180° S = 900° Thus the sum of the interior angle measure is 900°

Answer: S = (n – 2) . 180° S = (6 – 2) . 180° S = 4 . 180° S = 720° Thus the sum of the interior angle measure is 720°

Question 5. WRITING Explain how to find the sum of the interior measures of a polygon.

Answer: Steps to find the sum of the interior measurements of the polygon: 1. Count the number of sides of the polygon. 2. Subtract the number of sides by 2. 3. Multiply the result of the subtraction by 180°

Question 6. FINDING THE SUM OF INTERIOR ANGLE MEASURES Find the sum of the interior angle measures of the green polygon.

Answer: S = (n – 2) . 180° S = (4 – 2) . 180° S = 2 . 180° S = 360° Thus the sum of the interior angle measure is 360°

FINDING AN INTERIOR ANGLE MEASURE

Find the value of x.

Answer: S = (n – 2) . 180° S = (5 – 2) . 180° S = 3 . 180° S = 540° Thus the sum of the interior angle measure is 540° x° + 160° + 110° + 105° + 95° = 540° x° + 470° = 540° x° = 540° – 470° x° = 70° Thus the value of x is 70°.

Answer: S = (n – 2) . 180° S = (9 – 2) . 180° S = 7 . 180° S = 1260° Thus the sum of the interior angle measure is 1260° x° + 165° + 155° + 150° + 140° + 135° + 130° + 125° + 110° = 1260° x° + 1105° = 1260° x° = 1260° – 1105° x° = 155° Thus the value of x is 155°

Question 9. A company installs an octagonal swimming pool. a. Find the value of a for the pool shown at the left.

Answer: S = (n – 2) . 180° S = (8 – 2) . 180° S = 6 . 180° S = 1080° Thus the sum of the interior angle measure is 1080° a° + 120° + a° + 120° + a° + 120° + a° + 120° = 1080° 4a° + 480° = 1080° 4a° = 1080° – 480° 4a° = 600° a° = 600/4 a° = 150° Thus the value of x is 150°.

Answer: No for any octagon the sum of the interior angles is 1080 degrees.

Question 10. DIG DEEPER! A Bronze Star Medal A is shown. a. How many interior angles are there?

Answer: 10 interior angles are there

Answer: S = (n – 2) . 180° S = (10 – 2) . 180° S = 8 . 180° S = 1440° Thus the sum of the interior angle measure is 1440°

Answer: 60°

Explanation: Sum of all the angles = 180° x° + 60° + 60° = 180° x° + 120° = 180° x° = 180° – 120° x° = 60°

Answer: 45°

Explanation: Sum of all the angles = 180° x° + x° + 90° = 180° 2x° + 90° = 180° 2x° = 180° – 90° 2x° = 90° x° = 45°

Answer: 113°

Explanation: x° = 65° + 48° x° = 113° Thus the measure of an exterior angle is 113°

Solve the proportion.

Explanation: \(\frac{x}{12}\) = \(\frac{3}{4}\) 12 . \(\frac{x}{12}\) = \(\frac{3}{4}\) . 12 x = 3 . 3 x = 9

Explanation: \(\frac{14}{21}\) = \(\frac{x}{3}\) 3 . \(\frac{14}{21}\) = \(\frac{x}{3}\) . 3 x = 2

Explanation: \(\frac{9}{x}\) = \(\frac{6}{2}\) 2. \(\frac{9}{x}\) = 6 18 = 6x x = 3

Concepts, Skills, & Problem Solving EXPLORING INTERIOR ANGLES OF POLYGONS Use triangles to find the sum of the interior angle measures of the polygon. (See Exploration 1, p. 117.)

Answer: 360°

Explanation: Number of sides = 4 Number of interior triangles in the given figure = 2 The Sum of the measures of the interior angles using triangle = 2 . 180° = 360°

Answer: 1260°

Explanation: Number of sides = 9 Number of interior triangles in the given figure = 7 The Sum of the measures of the interior angles using triangle = 7 . 180° = 1260°

Answer: 540°

Explanation: Number of sides = 5 Number of interior triangles in the given figure = 3 The Sum of the measures of the interior angles using triangle = 3 . 180° = 540°

FINDING THE SUM OF INTERIOR ANGLE MEASURES Find the sum of the interior angle measures of the polygon.

Explanation: S = (n – 2) . 180° S = (4- 2) . 180° S = 2 . 180° S = 360° Thus the sum of the interior angle measure is 360°

Answer: 1080°

Explanation: S = (n – 2) . 180° S = (8- 2) . 180° S = 6 . 180° S = 1080° Thus the sum of the interior angle measure is 1080°

Explanation: S = (n – 2) . 180° S = (9- 2) . 180° S = 7 . 180° S = 1260° Thus the sum of the interior angle measure is 1260°

Answer: To find the sum of the interior angle measures he should subtract 2 from the number of sides of the polygon and then multiply by 180° S = (n – 2) . 180° By this, we can say that your friend is not correct.

FINDING AN INTERIOR ANGLE MEASURE Find the value of x.

Answer: S = (n – 2) . 180° S = (4- 2) . 180° S = 2 . 180° S = 360° Thus the sum of the interior angle measure is 360° x° + 155° + 25° + 137° = 360° x° + 317° = 360° x° = 360° – 317° x° = 43°

Answer: S = (n – 2) . 180° S = (6- 2) . 180° S = 4 . 180° S = 720° Thus the sum of the interior angle measure is 720° x° + x° + x° + x° + 90° + 90° = 720° 4x° + 180° = 720° 4x° = 720° – 180° 4x° = 540° x° = 540/4 x° = 135°

Answer: S = (n – 2) . 180° S = (6- 2) . 180° S = 4 . 180° S = 720° Thus the sum of the interior angle measure is 720° 3x° + 45° + 135° + x° + 135° + 45° = 720° 4x° + 360° = 720° 4x° = 720° – 360° 4x° = 360° x° = 360/4 x° = 90°

FINDING A MEASURE Find the measure of each interior angle of the regular polygon.

Answer: S = (n – 2) . 180° S = (3- 2) . 180° S = 1 . 180° S = 180° Thus the sum of the interior angle measure is 180° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 3. 180 ÷ 3 = 60°

S = (n – 2) . 180° S = (9 – 2) . 180° S = 7 . 180° S = 1260° Thus the sum of the interior angle measure is 1260° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 9. 1260 ÷ 9 = 140°

S = (n – 2) . 180° S = (12 – 2) . 180° S = 10 . 180° S = 1800° Thus the sum of the interior angle measure is 1800° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 12. 1800 ÷ 12 = 150°

Answer: No, my friend is not correct because to find the measure of each interior angle of a regular 20-gon, he should divide the sum of the measured interior angles by the number of interior angles, in this case, 20 but your friend divide it by 18 so he is not correct.

Answer: S = (n – 2) . 180° S = (5- 2) . 180° S = 3 . 180° S = 540° Thus the sum of the interior angle measure is 540° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 5. 540÷ 5 = 108°

b. RESEARCH Why are firehydrants made this way?

Question 22. PROBLEM SOLVING The interior angles of a regular polygon each measure 165°. How many sides does the polygon have?

Answer: (n – 2) . 180 = 165 . n 180n – 360 = 165n 180n – 360 + 360 – 165n = 165n + 360 – 165n 15n = 360 n = 360/15 n = 24 Therefore the polygon has 24 sides

Question 23. STRUCTURE A molecule can be represented by a polygon with interior angles that each measure 120°. What polygon represents the molecule? Does the polygon have to be regular? Justify your answers.

Answer: (n – 2) . 180 = 120 . n 180n – 360 = 120n 180n – 120n = 360 60n = 360 n = 6

Question 24. PROBLEM SOLVING The border of a Susan B. Anthony dollar is in the shape of a regular polygon. a. How many sides does the polygon have?

Answer: The polygon has 11 sides.

Answer: S = (n – 2) . 180° S = (11 – 2) . 180° S = 9 . 180° S = 1620° Thus the sum of the interior angle measure is 1620° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 11. 1620 ÷ 11 = 147°

Question 25. REASONING The center of the stained glass window is in the shape of a regular polygon. What are the measures of the interior angles of the green triangle?

Answer: (n-2)180°/n = (8-2)180°/8 = 135° m∠OAB = m∠OBA = 135/2 = 67.5° m∠AOB + m∠OAB + m∠OBA = 180° m∠AOB + 67.5° + 67.5° = 180° m∠AOB + 135° = 180° m∠AOB = 180° – 135° m∠AOB = 45°

Answer: The given polygon has 7 sides. S = (n – 2) . 180° S = (7 – 2) . 180° S = 5 . 180° S = 900° Thus the sum of the interior angle measure is 900° 4 . 135° + 3 . x° = 900° 540° + 3x° = 900° 3x° = 900° – 540° 3x° = 360° x° = 360/3 x° = 120°

Using Similar Triangles

Tell whether the triangles are similar. Explain.

Answer: Yes

Explanation: x° + 66° + 90° = 180° x° + 156° = 180° x° = 180° – 156° x° = 24° y° + 24° + 90° = 180° y° + 114° = 180° y° = 180° – 114° y° = 66° The triangles have two pairs of congruent angles. Thus the triangles are similar.

Explanation: We are not given any information about the lengths of the sides either, therefore with only a pair of congruent angles, we cannot tell whether the triangles are similar.

Explanation: x° + 54° + 63° = 180° x° + 107° = 180° x° = 180° – 107° x° = 63°

Answer: Option B

Explanation: ΔPQR and ΔTSR are congruent as TS || PQ leads to two pairs of correspondent congruent angles. ΔPQR is a dilation of ΔTSR because their sides are proportional, the constant of proportionality being greater than 1. ΔPQR is a scale drawing of ΔTSR because their sides are proportional. The question that does not fit is “Are ΔPQR and ΔTSR the same size and shape?” because the triangles do not have the same size, but they have the same shape.

Answer: Aqueduct/2.6 = 5/1 Aqueduct = 5 × 2.6 Aqueduct = 13 Thus the length of the Aqueduct is 13 km.

Answer: a/10 = 3/6 6 × a = 3 × 10 6a = 30 a = 30/6 a = 5 The length from point Z to point Y is 5 miles. Time to travel from point Z to point Y = 5/3.5 = 1.56 hour

Find the measure of each interior angle of the regular polygon.

Question 1. octagon

Answer: The measure of each interior angle is 135°

Explanation: S = (n – 2) . 180° S = (8- 2) . 180° S = 6 . 180° S = 1080° Thus the sum of the interior angle measure is 1080° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 8. 1080÷ 8= 135°

Question 2. decagon

Answer: The measure of each exterior angle is 144°

Explanation: S = (n – 2) . 180° S = (10 – 2) . 180° S = 8 . 180° S = 1440° Thus the sum of the interior angle measure is 1440° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 10. 1440÷ 10= 144°

Question 3. 18-gon

Answer: The measure of each interior angle is 160°

Explanation: S = (n – 2) . 180° S = (18- 2) . 180° S = 16 . 180° S = 2880° Thus the sum of the interior angle measure is 2880° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 18. 2880 ÷ 18= 160°

Solve the equation. Check your solution.

Question 4. 3.5 + y = -1

Answer: Given the equation 3.5 + y = -1 y = -1 – 3.5 y = -4.5

Question 5. 9x = 54

Answer: Given the equation 9x = 54 x = 54/9 x = 6

Question 6. -4 = \(\frac{2}{7}\)p

Answer: Given the equation -4 = \(\frac{2}{7}\)p -4 × 7 = 2p 2p = -28 p = -28/2 p = -14

Concepts, Skills, & Problem Solving CREATING SIMILAR TRIANGLES Draw a triangle that is either larger or smaller than the one given and has two of the same angle measures. Explain why the new triangle is similar to the original triangle. (See Exploration 1, p. 123.)

IDENTIFYING SIMILAR TRIANGLES Tell whether the triangles are similar. Explain.

Answer: The triangles have two pairs of congruent angles. So, the third angles are congruent, and the triangles are similar.

Answer: x° + 36° + 72° = 180° x° + 108° = 180° x° = 180° – 108° x° = 72° y° + 33° + 72° = 180° y° + 105° = 180° y° = 180° – 105° y° = 72° The triangles do not have two pairs of congruent angles. Therefore the triangles are not similar.

Answer: x° + 64° + 85° = 180° x° + 149° = 180° x° = 180° – 149° x° = 31° y° + 26° + 85° = 180° y° + 111° = 180° y° = 180° – 111° y° = 69° The triangles do not have two pairs of congruent angles. Therefore the triangles are not similar.

Answer: x° + 48° + 81° = 180° x° + 129° = 180° x° = 180° – 129° x° = 51° y° + 48° + 51° = 180° y° + 99° = 180° y° = 180° – 99° y° = 81° The triangles have two pairs of congruent angles. Therefore the triangles are similar.

Answer: 2x + 90 = 180° 2x = 180 – 90° 2x = 90° x = 90/2 x = 45° The ruler on the left and the ruler on the right both have the shape of a right triangle with 45° angles, therefore they are similar in shape, while the middle ruler has 60°, 30° angles.

STRUCTURE Tell whether the triangles are similar. Explain.

Answer: m∠APB + m∠B = 90° m∠APB + 51° = 90° m∠APB = 90° – 51° m∠APB = 39° m∠APB + m∠BPD + m∠DPC = 180° 39° + 102° + m∠DPC = 180° m∠DPC + 141° = 180 m∠DPC = 180 – 141° m∠DPC = 39° m∠A = m∠C m∠APB = m∠DPC

Answer: ∠APB ≅ ∠CPD m∠APB = m∠CPD m∠APB = 29° m∠A + m∠B + m∠APB = 180° m∠A + 88° + 29° = 180° m∠A  + 117° = 180° m∠A = 180° – 117° m∠A = 63° m∠PDC + m∠PDE = 180° m∠PDC + 91° = 180° m∠PDC = 180° – 91° m∠PDC = 89°

IDENTIFYING SIMILAR TRIANGLES Can you determine whether the triangles are similar? Explain.

Answer: PS || QR ∠PSQ and ∠SQR are interior angles using the transversal QS, thus they are congruent. ∠PSQ ≅ ∠SQR

Answer: As AB || DE there are two pairs of congruent alternate interior angles, using the transversals AE and BD. ∠A≅ ∠E ∠B≅ ∠D The two pairs of congruent angles are enough to prove that the triangles are similar. ΔABC ∼ ΔEDC

Answer: ΔAMN ∼ ΔABC MN/BC = AM/AB 1.5/d = 5/50 d = 1.5 × 10 d = 15 feet Therefore 15 feet is not an appropriate location.

Answer: The two triangles are similar because they are right triangles and ∠AXB ≅ ∠PXQ because they are vertical angles. PQ/300 = 80/240 240PQ = 24000 PQ = 24000/240 PQ = 100 steps

Answer: Given, A person who is 6 feet tall casts a 3-foot-long shadow. A nearby pine tree casts a 15-foot-long shadow. ΔXAB ∼ ΔXPQ AB/PQ = XB/XQ 6/PQ = 3/15 PQ = 30 ft

Question 21. OPEN-ENDED You place a mirror on the ground 6 feet from the lamppost. You move back 3 feet and see the top of the lamppost in the mirror. What is the height of the lamppost?

Question 22. DIG DEEPER! In each of two right triangles, one angle measure is two times another angle measure. Can you determine that the triangles are similar? Explain your reasoning.

Answer: We are given the right triangle ABC m∠A = 2m∠B Case 1: m∠A = 90° 90° = 2m∠B m∠B = 45° m∠C = 180° – 90° – 45° = 45° Case 2: m∠B = 90° m∠A = 2 × 90° = 180° Case 3: m∠C = 90° m∠A + m∠B = 180 – m∠C = 180° – 90° = 90° 2m∠B + m∠B = 90° 3m∠B = 90° m∠B = 30° m∠A = 2 . 30° = 60°

Answer: ΔABG ∼ ΔACF ΔACF ∼ ΔADE ΔABG ∼ ΔADE AB = BC = CD = BD/2 = 6.32/2 = 3.16 AB/CD = BG/DE 3BG = 6 BG = 2 feet ΔACF ∼ ΔADE AC/AD = CF/DE 2/3 = CF/6 3CF = 2(6) CF = 4 feet

Using the Problem-Solving Plan

Understand the problem You know two dimensions of a dog park and the ratio of the perimeter of the small dog section to the perimeter of the entire park. You are asked to find the area of each section. Make a plan Verify that the small triangle and the large triangle are similar. Then use the ratio of the perimeters to find the base or the height of each triangle and calculate the areas. Solve and check. Use the plan to solve the problem. Then check your solution.

Review Vocabulary

Choose and complete a graphic organizer to help you study the concept.

• interior angles formed by parallel lines and a transversal
• exterior angles formed by parallel lines and a transversal
• interior angles of a triangle
• exterior angles of a triangle
• similar triangles

Chapter Self-Assessment

3.1 Parallel Lines and Transversals (pp. 103–110)

Question 1. ∠8

Answer: 140°

Explanation: ∠8 and 140 degrees angle are alternate exterior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠8 is 140°

Question 2. ∠5

Explanation: ∠5 and 140 degrees angle are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠5 is 140°

Question 3. ∠7

Answer: 40°

Explanation: ∠5 and 140 degrees angle are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠5 is 140° ∠5 and ∠7 are supplementary angle. ∠5 + ∠7 = 180° 140° + ∠7 = 180° ∠7 = 180° – 140° ∠7 = 40° So, the measure of ∠7 is 40°

Question 4. ∠2

Explanation: 140 and ∠2 are supplementary angle. 140° + ∠2 = 180° ∠2 = 180° – 140° ∠2 = 40° So, the measure of ∠2 is 40°

Question 5. ∠6

Explanation: ∠5 and 140 degrees angle are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠5 is 140° ∠5 and ∠6 are supplementary angle. 140° + ∠6 = 180° ∠6 = 180° – 140° ∠6 = 40° So, the measure of ∠6 is 40°

Complete the statement. Explain your reasoning.

Answer: 123°

Explanation: ∠1 and ∠7 are alternate exterior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠7 is 123°

Answer: 122°

Explanation: ∠2 and ∠6 are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠6 is 58° ∠5 and ∠6 are supplementary angle. ∠5 + ∠6 = 180° 58° + ∠5 = 180° ∠5 = 180° – 58° ∠5 = 122° So, the measure of ∠5 is 122°

Answer: 119°

Explanation: ∠3 and ∠5 are alternate interior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠3 is 119°

Explanation: ∠4 and ∠6 are alternate exterior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠4 is 60°

Question 10. In Exercises 6–9, describe the relationship between ∠2 and ∠8.

Answer: ∠2 ≅ ∠8

Answer: ∠1 = 108°, ∠2 = 108°

Explanation: ∠3 and 72° are alternate interior angles. They are congruent. So, the measure of ∠3 is 72° ∠3 + ∠1 = 180° 72° + ∠1 = 180° ∠1 = 180° – 72° ∠1 = 108° So, the measure of ∠1 is 108° ∠1 and ∠2 are alternating interior angles. They are congruent.

3.2 Angles of Triangles (pp. 111 – 116)

Answer: Sum of all the angles in a triangle = 180° x° + 50° + 55° = 180° x° + 105° = 180° x° = 180° – 105° x° = 75°

Answer: Sum of all the angles in a triangle = 180° x° + (x + 8)° + 90° = 180° 2x° + 8° + 90° = 180° 2x° + 98° = 180° 2x° = 180° – 98° 2x° = 82 x° = 82/2 x° = 41° (x + 8)° = (41 + 8)° = 49°

Find the measure of the exterior angle.

Answer: s° = 50° + 75° s° = 125° Thus the measure of the exterior angle is 125°

Answer: Sum of all the angles in a triangle = 180° t° + (t + 10)° + (t + 20)° = 180° 3t° + 10° + 20° = 180° 3t° + 30° = 180° 3t° = 180° – 30° 3t° = 150° t° = 150/3 t° = 50° Exterior angle: t° + (t + 10)° t° + t° + 10° 2t° + 10° 2(50)° + 10° = 100° + 10° = 110° Thus the measure of the exterior angle is 110°.

Question 16. What is the measure of each interior angle of an equilateral triangle? Explain.

Answer: Sum of all the angles in a triangle = 180° x° + 30° + 56° = 180° x° + 86° = 180° x° = 180° – 86° x° = 94° Thus the measure of the interior angle of the triangle at Chertan = 94°

3.3 Angles of Polygons (pp. 117–122)

Find the sum of the interior angle measures of the polygon.

Answer: The polygon has 13 sides S = (n – 2) . 180° S = (13- 2) . 180° S = 11 . 180° S = 1980° Thus the sum of the interior angle measure is 1980°

Answer: The polygon has 9 sides S = (n – 2) . 180° S = (9- 2) . 180° S = 7 . 180° S = 1260° Thus the sum of the interior angle measure is 1260°

Answer: S = (n – 2) . 180° S = (4 – 2) . 180° S = 2 . 180° S = 360° Thus the sum of the interior angle measure is 360° x° + 60° + 128° + 95° = 360° x° + 283° = 360° x° = 360° – 283° x° = 77° Thus the value of x is 77°.

Answer: S = (n – 2) . 180° S = (7 – 2) . 180° S = 5 . 180° S = 900° Thus the sum of the interior angle measure is 900° x° + 135° + 125° + 135° + 105° + 150° + 140° = 900° x° + 790° = 900° x° = 900° – 790° x° = 110°

Answer: S = (n – 2) . 180° S = (6 – 2) . 180° S = 4 . 180° S = 720° Thus the sum of the interior angle measure is 720° x° + 120° + 140° + 92° + 125° + 130° = 720° x° + 607° = 720° x° = 720° – 607° x° = 113° The value of x° is 113°

Answer: The given polygon is an octagon. It has 8 sides. S = (n – 2) . 180° S = (8 – 2) . 180° S = 6 . 180° S = 1080° Thus the sum of the interior angle measure is 1080° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 8. 1080 ÷ 3 = 135°

3.4 Using Similar Triangles (pp. 123–128)

Answer: x° + 68° + 90° = 180° x° = 180° – 158° x° = 22° y° + 22° + 90° = 180° y° + 112° = 180° y° = 180° – 112° y° = 68° The triangles have two pairs of congruent angles. So, the triangles are similar.

Answer: x° + 100° + 30° = 180° x° + 130° = 180° x° = 180° – 130° x° = 50° y° + 100° + 50° = 180° y° + 150° = 180° y° = 180° – 150° y° = 30° The triangles have two pairs of congruent angles. So, the triangles are similar.

Answer: x° + 50° + 85° = 180° x° + 135° = 180° x° = 180° – 135° x° = 45° y° + 85° + 35° = 180° y° + 120° = 180° y° = 180° – 120° y° = 60° The triangles do not have two pairs of congruent angles. So, the triangles are not similar.

Answer: ∠B ≅ ∠D ∠A ≅ ∠C ∠AXB ≅ ∠CXD ∠AXB and ∠CXD are vertical angles. ΔAXB ∼ ΔCXD

Question 28. A person who is 5 feet tall casts a shadow that is 4 feet long. A nearby building casts a shadow that is 24 feet long. What is the height of the building?

Answer: Given, A person who is 5 feet tall casts a shadow that is 4 feet long. A nearby building casts a shadow that is 24 feet long. Let the height of the building = x ft x/24 = 5/4 24 . x/24 = 5/4 . 24 x = 30 Thus the height of the building is 30 ft.

Practice Test

Question 1. ∠7

Answer: 47°

Explanation: ∠7 and 47° angles are alternate exterior angles formed by a transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠7 is 47°

Explanation: ∠6 and 47° angles are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠6 is 47°

Answer: 133°

Explanation: ∠4 and 47° are supplementary angles. 47° + ∠4 = 180° ∠4 = 180° – 47° ∠4 = 133° Thus the measure of ∠4 = 133°

Question 4. ∠5

Explanation: ∠6 and 47° angles are corresponding angles formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠6 is 47° ∠6 + ∠5 = 180° 47° + ∠5 = 180° ∠5 = 180° – 47° ∠5 = 133° Thus the measure of ∠5 = 133°

Answer: 28°

Explanation: Sum of all the angles in a triangle = 180° x° + 129° + 23° = 180° x° + 152° = 180° x° = 180° – 152° x° = 28° Thus the value of x° is 28°

Answer: 68°

Explanation: Sum of all the angles in a triangle = 180° x° + (x – 24)° + 68° = 180° x° + x° – 24° + 68° = 180° 2x° + 44° = 180° 2x° = 180° – 44° 2x° = 136° x° = 68° (x – 24)° = (68 – 24)° = 44°

Answer: j° = 40° + 90° j° = 130° The measure of an exterior angle is 130°.

Answer: The coin has 7 sides. S = (n – 2) . 180° S = (7 – 2) . 180° S = 5 . 180° S = 900° Thus the sum of the interior angle measure is 900°

Answer: S = (n – 2) . 180° S = (5 – 2) . 180° S = 3 . 180° S = 540° Thus the sum of the interior angle measure is 540° 2x° + 125° + 90° + 2x° + 125° = 540° 4x° + 340° = 540° 4x° = 540° – 340° 4x° = 200° x° = 200/4 x° = 50° The value of x° is 50°

Answer: S = (n – 2) . 180° S = (6 – 2) . 180° S = 4 . 180° S = 720° Thus the sum of the interior angle measure is 720° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 6. 720 ÷ 6 = 120°

Answer: To find x°: x° + 61° + 70° = 180° x° + 131° = 180° x° = 180° – 131° x° = 49° To find y°: x° + 39° + 70° = 180° x° + 109° = 180° x° = 180° – 109° x° = 71° The triangles do not have two pairs of congruent angles. So, the triangles are not similar.

Answer: ∠A ≅ ∠QPB ∠C ≅ ∠PQB ΔBPQ ∼ ΔBAC

Answer: One way: ∠3 and 65° are supplementary angles. ∠5 and ∠3 are alternate interior angles. Another way: ∠8 and 65° are alternate exterior angles. ∠5 and ∠8 are supplementary angles.

Answer: Given, You swim 3.6 kilometers per hour. d/105 = 80/140 105 . d/105 = 80/140 . 105 d = 60 The length of the pond is 60 m. Speed = 3.6 km per hour = 1 m sec Distance = d = 60m Time it will take to swim across the pond = distance/speed = 60/1 = 60 sec = 1 min

Answer: C = 11 + 1.6t C – 11 = 1.6t 1.6t = C – 11 t = (C – 11)/1.6 Thus the correct answer is option B.

Answer: 5(x – 4) = 3x 5x – 20 = 3x 5x – 3x = 20 2x = 20 x = 20/2 x = 10 Thus the correct answer is option I.

Answer: △PQR is similar to △STU PQ = 12 ST = 16 SU = 20 TU = 18 PQ/ST = QR/TU 12/16 = X/18 16X = 12 × 18 X = 216/16 X = 13.5 cm Thus the correct answer is option C.

Answer: ∠y and 125° are supplementary angles. 125° + ∠y = 180° ∠y = 180° – 125° ∠y = 55° So, the measure of ∠y = 55° ∠x and ∠y are alternate interior angles. They are congruent. So, the measure of ∠x = 55°

Answer: My friend made the error by multiplying both sides by –\(\frac{2}{5}\). To correct the error she should multiply both sides by –\(\frac{5}{2}\) instead of –\(\frac{2}{5}\) Thus the correct answer is option F.

Answer: Given, X(-6,-1) Y(-3,-5) X(-2,-3) Reflecting a point (x,y) in the y-axis. (x, y) = (-x, y) X(-6,-1) = X'(6, -1) Y(-3,-5) = Y'(3, -5) X(-2,-3) = Z'(2, -3) Thus the correct answer is option B.

Answer: S = (n – 2) . 180° Part B A quadrilateral has angles measuring 100°, 90°, and 90°. Find the measure of its fourth angle. Show your work and explain your reasoning.

Answer: The quadrilateral has 4 sides S = (n – 2) . 180° S = (4 – 2) . 180° S = 2 . 180° S = 360 ° Thus the sum of the interior angles is 360 ° x° + 100° + 90° + 90° = 360° x° + 280° = 360° x° = 360° – 280° x° = 80° Thus the value of x° is 80°

Answer: Number of sides = 3 The number of interior triangles in the given figure = 3 Sum of the interior angles measure using triangle = 3 × 180° = 540

Conclusion:

I wish the details prevailed in the above article is beneficial for all the 8th grade students. Hope our Big Ideas Math Answers Grade 8 Chapter 3 Angles and Triangles helped you a lot to overcome the difficulties in this chapter. Feel free to post your comments in the comment box. Stay tuned to our ccssmathanswers.com to get step by step explanation for all the Grade 8 chapters.

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1.1: Similar Triangles

• Last updated
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• Page ID 112401

• Katherine Yoshiwara
• Los Angeles Pierce College

Congruent Triangles

Two triangles are congruent if they have exactly the same size and shape. This means that their corresponding angles are equal, and their corresponding sides have the same lengths, as shown below.

Example 1.17

The two triangles below are congruent. List the corresponding parts, and find the angles \(\theta\), \(\phi\), and \(\chi\), and side \(z\).

In these triangles, \(\angle B=\angle D\) because they are both right angles, and \(\angle B C A= \angle D C E\) because they are vertical angles, so \(\theta=25^{\circ}\). The third angles, \(\angle A\) and \(\angle E\), must also be equal, so \(\phi=\chi=65^{\circ}\). (Do you see why?) The sides opposite each pair of corresponding angles are equal, so \(A B=D E, B C=C D\) and \(A C=C E\). In particular, we find that \(z=9\).

Checkpoint 1.18

The two triangles at right are congruent. Find the values of \(\alpha, \beta\), and \(\gamma\).

\( \alpha=85^{\circ}, \beta=75^{\circ}, \gamma=85^{\circ}\)

Recall that the altitude of a triangle is the segment from one vertex of the triangle perpendicular to the opposite side.

Example 1.19

Show that the altitude of an equilateral triangle divides it into two congruent right triangles.

Consider for example an equilateral triangle of side 8 inches, as shown above. The altitude is perpendicular to the base, so each half of the original triangle is a right triangle. Because each right triangle contains a \(60^{\circ}\) angle, the remaining angle in each triangle must be \(90^{\circ}-60^{\circ}=30^{\circ}\). Both triangles have a side of length 8 between the angles of \(30^{\circ}\) and \(60^{\circ}\), so they are congruent. (Consequently, the short sides of the congruent triangles are equal, so each is half the original base.)

The triangles in the previous example are a special type of right triangle called \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles. Notice that in these triangles, the leg opposite the \(30^{\circ}\) angle is half the length of the hypotenuse .

Checkpoint 1.20

The diagonal of a parallelogram divides it into two congruent triangles, as shown at right. List the corresponding parts of the two triangles, and explain why each pair is equal.

\(\angle B C A=\angle C A D\) and \(\angle B A C=\angle A C D\) because they are alternate interior angles. If two pairs of angles in a triangle are equal, so is the third pair, so \(\angle B=\angle D\). \(B C=A D\) and \(A B=C D\) because they are opposite sides of a parallelogram, and \(A C=A C\).

Similar Triangles

Two triangles are similar if they have the same shape but not necessarily the same size. The corresponding angles are equal, and the corresponding sides are proportional . We can think of one similar triangle as an enlargement or a reduction of the other. (See the figures below.)

To decide whether two triangles are similar, it turns out that we need to verify only one of the two conditions for similarity, and the other condition will be true automatically.

Similar Triangles.

Two triangles are similar if either

• their corresponding angles are equal, or
• their corresponding sides are proportional.

Example 1.21

Which of the pairs of triangles shown below are similar?

a. We will check whether the corresponding sides are proportional. We compute the ratios of the corresponding sides, making sure to write each ratio in the same order, \(\frac{\text { larger triangle }}{\text { smaller triangle }}\). (The other order, \(\frac{\text { smaller triangle }}{\text { larger triangle }}\), would also work, as long as we use the same order for all the ratios.)

shorter legs: \(\dfrac{9}{6} \quad\) longer legs: \(\dfrac{12}{8} \quad\) hypotenuses: \(\dfrac{15}{10}\)

Because all of these ratios are equal to 1.5, the triangles are similar.

b. The ratios of corresponding sides are not equal: the ratio of the longest sides is \(\frac{6}{5}\), but the ratio of the smallest sides is \(\frac{4}{3}\). The triangles are not similar.

c. The missing angle of the first right triangle is \(48^{\circ}\), and the missing angle in the second right triangle is \(42^{\circ}\), so three pairs of angles match. The triangles are similar.

Checkpoint 1.22

Are the triangles below similar? Explain why or why not in each case.

a. The triangles are similar because \(\frac{4}{6} = \frac{6}{9} = \frac{8}{12}\), so the sides are proportional.

b. The third angle in both triangles is \(80^{\circ}\), so the triangles are similar because their corresponding angles are equal.

Note 1.23 In part (b) of the previous exercise, note that the third angle in each triangle must be \(80^{\circ}\), because the sum of the angles is \(180^{\circ}\). Thus, we need only show that two pairs of angles are equal to show that two triangles are similar.

Using Proportions with Similar Triangles

The figure in the next example shows a parallelogram \(A B C D\) and two triangles, \(\triangle A B E\) and \(\triangle F C E\). Can we find the unknown lengths \(x\) and \(y\) in the larger triangle?

First note that two pairs of corresponding angles in the triangles are equal: \(\angle B E A\) and \(\angle F E C\) are vertical angles, and \(\angle E F C\) and \(\angle B A E\) are alternate interior angles. But if two pairs of corresponding angles are equal, then the third pair must be equal also. This means that the two triangles are similar, and we can use the fact that their corresponding sides are proportional to find \(x\) and \(y\).

Example 1.24

Find the value of \(x\) in the figure at right.

We see that \(x\) is the length of the shortest side in \(\triangle A B E\). We know the short side in \(\triangle F C E\) and the lengths of the medium sides in each triangle. If we form the ratios of the short sides and the medium sides, we obtain the following proportion.

\[\dfrac{\text { larger triangle }}{\text { smaller triangle }}: \dfrac{x}{4}=\dfrac{15}{6} \nonumber\]

To solve the proportion, we cross-multiply to get

\begin{aligned} 6 x &=4(15)=60 \quad \quad \quad \quad \text{Divide both sides by 6.} \\ x &=\dfrac{60}{6}=10 \end{aligned}

Caution 1.25

Remember that “cross-multiplying” is a short-cut technique for solving proportions. It does not work for other operations involving fractions.

Checkpoint 1.26

Find the value of \(y\) in the previous example.

Similar Right Triangles

If two right triangles have one pair of corresponding acute angles with the same measure, then the triangles are similar. We can use this fact about right triangles to make indirect measurements.

Example 1.27

Delbert would like to know the height of a certain building. He gets Francine to hold up a 5-foot pole near the building and measures the length of its shadow. The shadow of the pole is 3 feet long, and the shadow of the building is 12 feet long.

a. Use similar triangles to write a proportion involving the height of the building.

b. Solve the proportion to find the height of the building.

a. In the figure above we see two right triangles: One triangle is formed by the building and its shadow, and the other by the pole and its shadow. Because the light rays from the sun are parallel, the two angles at the tips of the shadows are equal. Thus, the two right triangles are similar, and their corresponding sides are proportional. The ratios of heights and bases in the two triangles yield the proportion

\[\dfrac{\text{larger triangle}}{\text{smaller triangle}}: \dfrac{h}{5} = \dfrac{12}{3}\]

b. To solve the proportion, we cross-multiply to get

\begin{aligned} 3h& = 5(12) = 60 \quad \quad \quad \quad \text{Divide both sides by 3.} \\ h&=\dfrac{60}{3} = 20 \end{aligned}

The building is 20 feet tall.

Checkpoint 1.28

Earlier we created a \(30°-60°-90°\) triangle in which the shorter leg was 4 inches and the hypotenuse was 8 inches. The hypotenuse of another \(30°-60°-90°\) triangle is 5 feet. What is the length of the side opposite the \(30°\) angle?

\(2.5\) feet

Overlapping Triangles

In some applications, similar triangles may share a side or an angle.

Example 1.29

Identify two similar triangles in the figure at right, and write a proportion to find \(H\).

The two triangles overlap, sharing the marked angle, as shown below. Because each triangle also has a right angle, they are similar.

Note that the base of the larger triangle is \(24 + 12 = 36\). The ratio of the heights and the ratio of the bases must be equal, so we write the following proportion.

\begin{aligned} \dfrac{H}{10} &= \dfrac{36}{24} \quad \quad \quad \quad &&\text{Cross-multiply.} \\ 24H &= 360 &&\text{Divide both sides by 24.} \\ H &= \dfrac{360}{24} = 15 \end{aligned}

Checkpoint 1.30

Heather wants to know the height of a street lamp. She discovers that when she is 12 feet from the lamp, her shadow is 6 feet long. Find the height of the street lamp.

\(15\) feet

Review the following skills you will need for this section.

Algebra Refresher 1.3

Which of the following expressions and equations are proportions?

1. \(\dfrac{7}{x} = \dfrac{3}{5}\)

2. \(\dfrac{x}{2} = \dfrac{8}{x+2}\)

3. \(1+\dfrac{x}{4} = \dfrac{2x}{3}\)

4. \(\dfrac{6}{x} + \dfrac{x}{5}\)

5. \(\dfrac{3}{x+1} \cdot \dfrac{2x}{5}\)

6. \(\dfrac{1}{x} + \dfrac{2}{3x} = \dfrac{x-2}{2}\)

Solve each equation. Begin by writing an equivalent equation without fractions: multiply both sides by the LCD.

7. \(\dfrac{x}{12} = \dfrac{3}{x}\)

8. \(1+\dfrac{x}{2} = \dfrac{2x}{5}\)

Only 1 and 2 are proportions.

7. \(\pm6\)

Section 1.2 Summary

Look up the definitions of new terms in the Glossary.

• Congruent

• Altitude

• Hypotenuse

• Parallelogram

• Similar

• Proportional

1. Two triangles are congruent if they have exactly the same size and shape.

2. The altitude of an equilateral triangle divides it into two congruent right triangles.

3. In a \(30^{\circ}-60^{\circ}-90^{\circ}\) right triangle, the leg opposite the \(30^{\circ}\) angle is half the length of the hypotenuse .

4. Two triangles are similar if they have the same shape but not necessarily the same size. The corresponding angles are equal, and the corresponding sides are proportional .

5. Two triangles are similar if either

6. If two right triangles have one pair of corresponding acute angles with the same measure, then the triangles are similar.

Study Questions

1 What is the difference between congruent triangles and similar triangles?

2 What is the name of the short-cut method for solving proportions? Why does the method work?

3 In two triangles, if two corresponding pairs of angles are equal, are the triangles similar? How do you know?

4 For the triangles shown, which of the following equations is true? Explain why.

a. \(\dfrac{4}{x} = \dfrac{6}{8}\)

b. \(\dfrac{x}{4} = \dfrac{6}{8}\)

c. \(\dfrac{x}{x+4} = \dfrac{6}{8}\)

d. \(\dfrac{x}{x+4} = \dfrac{6}{14}\)

Practice each skill in the Homework Problems listed.

1 Identify congruent triangles and find unknown parts #1-6

2 Identify similar triangles #7-10

3 Find unknown parts of similar triangles #11-20

4 Solve problems using proportions and similar triangles #21-26

5 Use proportions to relate sides of similar triangles #27-38

Homework 1.2

In Problems 1–4, name two congruent triangles and find the unknown quantities.

1. \(P QRS\) is an isosceles trapezoid.

3. \(\Delta PRU\) is isosceles.

4. \(\Delta P RU\) is isosceles and \(OR = NG\). Find \(\angle RNG\) and \(\angle RNO\).

5. Delbert and Francine want to measure the distance across a stream. They mark point \(A\) directly across the stream from a tree at point \(T\) on the opposite bank. Delbert walks from point \(A\) down the bank a short distance to point \(B\) and sights the tree. He measures the angle between his line of sight and the streambank.

a Draw a figure showing the stream, the tree, and right triangle \(ABT\).

b Meanwhile Francine, who was still standing at point \(A\), walks away from the stream at right-angles to Delbert’s path. Delbert watches her progress, and tells her to stop at point \(C\) when the angle between the stream bank and his line of sight to Francine is the same as the angle from the stream bank to the tree. Add triangle \(ABC\) to your figure.

c Delbert now measures the distance from point \(A\) to Francine at point \(C\). Explain why this distance is the same as the distance across the stream.

6. If you have a baseball cap, here is another way to measure the distance across a river. Stand at point \(A\) directly across the river from a convenient landmark, say a large rock, on the other side. Tilt your head down so that the brim of the cap points directly at the base of the rock, \(R\).

a Draw a figure showing the river, the rock, and right triangle \(ABR\), where \(B\) is the location of your baseball cap on your head.

b Now, without changing the angle of your head, rotate \(90^{\circ}\) and sight along the bank on your side of the river. Have a friend mark the spot \(C\) on the ground where the brim of your cap points. Add triangle \(ABC\) to your figure.

c Finally, you can measure the distance from point \(A\) to point \(C\). Explain why this distance is the same as the distance across the river.

For Problems 7–10, decide whether the triangles are similar, and explain why or why not.

Assume the triangles in Problems 11–14 are similar. Solve for the variables. (Figures are not drawn to scale.)

In Problems 15–20, use properties of similar triangles to solve for the variable.

For Problems 21–26, use properties of similar triangles to solve.

21. A rock climber estimates the height of a cliff she plans to scale as follows. She places a mirror on the ground so that she can just see the top of the cliff in the mirror while she stands straight. The angles 1 and 2 formed by the light rays are equal, as shown in the figure. She then measures the distance to the mirror (2 feet) and the distance from the mirror to the base of the cliff (56 feet). If she is 5 feet 6 inches tall, how high is the cliff?

22. Edo estimates the height of the Washington Monument as follows. He notices that he can see the reflection of the top of the monument in the reflecting pool. He is feet from the tip of the reflection, and that point is 1080 yards from the base of the monument, as shown below. From his physics class Edo knows that the angles marked and are equal. If Edo is 6 feet tall, what is his estimate for the height of the Washington Monument?

23. In the sixth century BC, the Greek philosopher and mathematician Thales used similar triangles to measure the distance to a ship at sea. Two observers on the shore at points \(A\) and \(B\) would sight the ship and measure the angles formed, as shown in figure (a). They would then construct a similar triangle as shown in figure (b), with the same angles at \(A\) and \(B\), and measure its sides. (This method is called triangulation .) Use the lengths given in the figures to find the distance from observer to the ship.

24. The Capilano Suspension Bridge is a footbridge that spans a 230-foot gorge north of Vancouver, British Columbia. Before crossing the bridge, you decide to estimate its length. You walk 100 feet downstream from the bridge and sight its far end, noting the angle formed by your line of sight, as shown in figure (a). You then construct a similar right triangle with a two-centimeter base, as shown in figure (b). You find that the height of your triangle is 8.98 centimeters. How long is the Capilano Suspension Bridge?

25. A conical tank is 12 feet deep and the diameter of the top is 8 feet. If the tank is filled with water to a depth of 7 feet as shown in the figure at right, what is the area of the exposed surface of the water?

26. To measure the distance \(EC\) across the lake shown in the figure at right, stand at \(A\) and sight point \(C\) across the lake, then mark point \(B\). Then sight to point \(E\) and mark point \(D\) so that \(DB\) is parallel to \(EC\). If \(AD = 25\) yards, \(AE = 60\) yards, and \(BD = 30\) yards, how wide is the lake?

In Problems 27–28, the pairs of triangles are similar. Solve for \(y\) in terms of \(x\). (The figures are not drawn to scale.)

For Problems 29–34, use properties of similar triangles to solve for the variable.

In Problems 35–38, solve for \(y\) in terms of \(x\)

39. Triangle \(ABC\) is a right triangle, and \(AD\) meets the hypotenuse \(BC\) at a right angle.

a If \(\angle ACB = 20^{\circ}\), find \(\angle B, \angle CAD\), and \(\angle DAB\).

b Find two triangles similar to \(\Delta ABC\). List the corresponding sides in each of the triangles.

40. Here is a way to find the distance across a gorge using a carpenter’s square and a five-foot pole. Plant the pole vertically on one side of the gorge at point \(A\) and place the angle of the carpenter’s square on top of the pole at point \(B\), as shown in the figure. Sight along one side of the square so that it points to the opposite side of the gorge at point \(P\). Without moving the square, sight along the other side and mark point \(Q\). If the distance from \(Q\) to \(A\) is six inches, calculate the width of the gorge. Explain your method.