angle relationships homework 2 applying angle relationships

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Applying Angle Relationships HW 2

Mathematics.

14 questions

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The following angles are which type?

Alternate Interior

Corresponding

Same Side Interior

Alternate Exterior

What is the equation to solve for x?

3x+21=6x-60

3x+21+6x-60=180

What is the angle measurement?

What is the equation for the angle relationship?

What is the angle pair?

vertical angles

alternate interior angles

supplementary angles

corresponding angles

What is the measurement of the angles?

What is the relationship between the two angles?

supplementary

corresponding

What is the equation for the angle pair?

What are the angle measurements of the angles?

180 and 180

What is the relationship of the angle pair?

The angles equal 100 degrees

The angles are congruent

The angles equal 180

The angles are equal each other

3x+10=3x-28

What is the relationship between the angle pairs?

angles are congruent

angles equal 5 degrees

angles are supplementary

angles are on same side of the transversal

What is the equation of the angle pairs?

15x+12x+15=180

What is the angle measures of the angle pairs

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2.0: Side and Angle Relationships

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• Page ID 112415

• Katherine Yoshiwara
• Los Angeles Pierce College

Introduction

From geometry we know that the sum of the angles in a triangle is \(180^{\circ}\). Are there any relationships between the angles of a triangle and its sides?

First of all, you have probably observed that the longest side in a triangle is always opposite the largest angle, and the shortest side is opposite the smallest angle, as illustrated below.

It is usual to label the angles of a triangle with capital letters, and the side opposite each angle with the corresponding lower-case letter, as shown at right. We will follow this practice unless indicated otherwise.

Example 2.2

In \(\triangle F G H, \angle F=48^{\circ}\), and \(\angle G\) is obtuse. Side \(f\) is 6 feet long. What can you conclude about the other sides?

Because \(\angle G\) is greater than \(90^{\circ}\), we know that \(\angle F+\angle G\) is greater than \(90^{\circ}+48^{\circ}=138^{\circ}\), so \(\angle F\) is less than \(180^{\circ}-138^{\circ}=42^{\circ}\). Thus, \(\angle H<\angle F<\angle G\), and consequently \(h<f<g\). We can conclude that \(h<6\) feet long, and \(g>6\) feet long.

Checkpoint 2.3

In isosceles triangle \(\triangle R S T\), the vertex angle \(\angle S=72^{\circ}\). Which side is longer, \(s\) or \(t\)?

\(s\) is longer

The Triangle Inequality

It is also true that the sum of the lengths of any two sides of a triangle must be greater than the third side, or else the two sides will not meet to form a triangle. This fact is called the triangle inequality.

Triangle Inequality.

In any triangle, we must have that

\(p + q > r\)

where \(p, q\), and \(r\) are the lengths of the sides of the triangle.

We cannot use the triangle inequality to find the exact lengths of the sides of a triangle, but we can find largest and smallest possible values for the length.

Example 2.4

Two sides of a triangle have lengths 7 inches and 10 inches, as shown at right. What can you say about the length of the third side?

We let \(x\) represent the length of the third side of the triangle. By looking at each side in turn, we can apply the triangle inequality three different ways, to get

\[7<x+10, \quad 10<x+7, \quad \text { and } \quad x<10+7 \nonumber\]

We solve each of these inequalities to find

\[-3<x, \quad 3<x, \quad \text { and } \quad x<17 \nonumber\]

We already know that \(x>-3\) because \(x\) must be positive, but the other two inequalities do give us new information. The third side must be greater than 3 inches but less than 17 inches long.

Checkpoint 2.5

Can you make a triangle with three wooden sticks of lengths 14 feet, 26 feet, and 10 feet? Sketch a picture, and explain why or why not.

No, \(10+14\) is not greater than 26 .

Right Triangles: The Pythagorean Theorem

In Chapter 1 we used the Pythagorean theorem to derive the distance formula. We can also use the Pythagorean theorem to find one side of a right triangle if we know the other two sides.

Pythagorean Theorem

In a right triangle, if \(c\) stands for the length of the hypotenuse, and the lengths of the two legs are denoted by \(a\) and \(b\), then

\(a^2 + b^2 = c^2\)

Example 2.6

A 25 -foot ladder is placed against a wall so that its foot is 7 feet from the base of the wall. How far up the wall does the ladder reach?

We make a sketch of the situation, as shown below, and label any known dimensions. We'll call the unknown height \(h\).

The ladder forms the hypotenuse of a right triangle, so we can apply the Pythagorean theorem, substituting 25 for \(c, 7\) for \(b\), and \(h\) for \(a\).

\begin{gathered} a^2+b^2=c^2 \\ h^2+7^2=25^2 \end{gathered}

Now solve by extraction of roots:

\begin{aligned} h^2+49 &=625 &&\text{Subtract 49 from both sides.} \\ h^2 &=576 &&\text{Extract roots.}\\ h &=\pm \sqrt{576} &&\text{Simplify the radical.}\\ h &=\pm 24 \end{aligned}

The height must be a positive number, so the solution \(-24\) does not make sense for this problem. The ladder reaches 24 feet up the wall.

Checkpoint 2.7

A baseball diamond is a square whose sides are 90 feet long. The catcher at home plate sees a runner on first trying to steal second base, and throws the ball to the second-baseman. Find the straight-line distance from home plate to second base.

\(90 \sqrt{2} \approx 127.3\) feet

Keep in mind that the Pythagorean theorem is true only for right triangles, so the converse of the theorem is also true. In other words, if the sides of a triangle satisfy the relationship \(a^2+b^2=c^2\), then the triangle must be a right triangle. We can use this fact to test whether or not a given triangle has a right angle.

Example 2.9

Delbert is paving a patio in his back yard, and would like to know if the corner at \(C\) is a right angle.

He measures 20 cm along one side from the corner, and 48 cm along the other side, placing pegs \(P\) and \(Q\) at each position, as shown at right. The line joining those two pegs is 52 cm long. Is the corner a right angle?

If is a right triangle, then its sides must satisfy \(p^2+q^2=c^2\). We find

\begin{aligned} p^2+q^2 &=20^2+48^2=400+2304=2704 \\ c^2 &=52^2=2704 \end{aligned}

Yes, because \(p^2+q^2=c^2\), the corner at \(C\) is a right angle.

Checkpoint 2.10

The sides of a triangle measure 15 inches, 25 inches, and 30 inches long. Is the triangle a right triangle?

The Pythagorean theorem relates the sides of right triangles. However, for information about the sides of other triangles, the best we can do (without trigonometry!) is the triangle inequality. Nor does the Pythagorean theorem help us find the angles in a triangle. In the next section we discover relationships between the angles and the sides of a right triangle.

Review the following skills you will need for this section.

Algebra Refresher 2.2

Solve the inequality.

1. \(6-x>3\)

2. \(\dfrac{-3 x}{4} \geq-6\)

3. \(3 x-7 \leq-10\)

4. \(4-3 x<2 x+9\)

If \(x<0\), which of the following expressions are positive, and which are negative?

6. \(-(-x)\)

8. \(-|x|\)

9. \(-|-x|\)

10. \(x^{-1}\)

1 \(x<3\)

2 \(x \leq 8\)

3 \(x \leq -1\)

4 \(x > -1\)

10 Negative

Section 2.1 Summary

• Converse • Extraction of roots • Inequality

1 The longest side in a triangle is opposite the largest angle, and the shortest side is opposite the smallest angle.

2 Triangle Inequality: In any triangle, the sum of the lengths of any two sides is greater than the length of the third side.

3 Pythagorean Theorem: In a right triangle with hypotenuse \(c\), \(a^2 + b^2 = c^2\).

4 If the sides of a triangle satisfy the relationship \(a^2 + b^2 = c^2\), then the triangle is a right triangle.

Study Questions

1 Is it always true that the hypotenuse is the longest side in a right triangle? Why or why not?

2 In \(\triangle D E F\), is it possible that \(d+e>f\) and \(e+f>d\) are both true? Explain your answer.

3 In a right triangle with hypotenuse \(c\), we know that \(a^2+b^2=c^2\). Is it also true that \(a+b=c\)? Why or why not?

4 The two shorter sides of an obtuse triangle are 3 in and 4 in. What are the possible lengths for the third side?

1 Identify inconsistencies in figures #1-12

2 Use the triangle inequality to put bounds on the lengths of sides #13-16

3 Use the Pythagorean theorem to find the sides of a right triangle #17-26

4 Use the Pythagorean theorem to identify right triangles #27-32

5 Solve problems using the Pythagorean theorem #33-42

Homework 2.1

For Problems 1–12, explain why the measurements shown cannot be accurate.

13. If two sides of a triangle are 6 feet and 10 feet long, what are the largest and smallest possible values for the length of the third side?

14. Two adjacent sides of a parallelogram are 3 cm and 4 cm long. What are the largest and smallest possible values for the length of the diagonal?

15. If one of the equal sides of an isosceles triangle is 8 millimeters long, what are the largest and smallest possible values for the length of the base?

16. The town of Madison is 15 miles from Newton, and 20 miles from Lewis. What are the possible values for the distance from Lewis to Newton?

For Problems 17–22,

a Make a sketch of the situation described, and label a right triangle.

b Use the Pythagorean Theorem to solve each problem.

17. The size of a TV screen is the length of its diagonal. If the width of a 35-inch TV screen is 28 inches, what is its height?

18. If a 30-meter pine tree casts a shadow of 30 meters, how far is the tip of the shadow from the top of the tree?

19. The diagonal of a square is 12 inches long. How long is the side of the square?

20. The length of a rectangle is twice its width, and its diagonal is meters long. Find the dimensions of the rectangle.

21. What size rectangle can be inscribed in a circle of radius 30 feet if the length of the rectangle must be three times its width?

22. What size square can be inscribed inside a circle of radius 8 inches, so that its vertices just touch the circle?

For Problems 23–26, find the unknown side of the triangle.

For Problems 27–32, decide whether a triangle with the given sides is a right triangle.

27. 9 in, 16 in, 25 in

28. 12 m, 16 m, 20 m

29. 5 m, 12 m, 13 m

30. 5 ft, 8 ft, 13 ft

31. \(5^2\) ft, \(8^2\) ft, \(13^2\) ft

32. \(\sqrt{5}\) ft, \(\sqrt{8}\) ft, \(\sqrt{13}\) ft

33. Show that the triangle with vertices (0, 0), (6, 0) and (3, 3) is an isosceles right triangle, that is, a right triangle with two sides of the same length.

34. Two opposite vertices of a square are \(A(−9, −5)\) and \(C(3, 3)\).

a Find the length of a diagonal of the square.

b Find the length of the side of the square.

35. A 24-foot flagpole is being raised by a rope and pulley, as shown in the figure. The loose end of the rope can be secured to a ring on the ground 7 feet from the base of the pole. From the ring to the top of the pulley, how long should the rope be when the flagpole is vertical?

36. To check whether the corners of a frame are square, carpenters sometimes measure the sides of a triangle, with two sides meeting at the join of the boards. Is the corner shown in the figure square?

37. Find \(\alpha, \beta\) and \(h\).

38. Find \(\alpha, \beta\) and \(d\).

39. Find the diagonal of a cube of side 8 inches. Hint: Find the diagonal of the base first.

40. Find the diagonal of a rectangular box whose sides are 6 cm by 8 cm by 10 cm. Hint: Find the diagonal of the base first.

For Problems 41 and 42, make a sketch and solve.

a The back of Brian’s pickup truck is five feet wide and seven feet long. He wants to bring home a 9-foot length of copper pipe. Will it lie flat on the floor of the truck?

42. What is the longest curtain rod that will fit inside a box 60 inches long by 10 inches wide by 4 inches tall?

43. In this problem, we’ll show that any angle inscribed in a semi-circle must be a right angle. The figure shows a triangle inscribed in a unit circle, one side lying on the diameter of the circle and the opposite vertex at point \((p, q)\) on the circle.

a What are the coordinates of the other two vertices of the triangle? What is the length of the side joining those vertices?

b Use the distance formula to compute the lengths of the other two sides of the triangle.

c Show that the sides of the triangle satisfy the Pythagorean theorem, \(a^2 + b^2 = c^2\).

44. There are many proofs of the Pythagorean theorem. Here is a simple visual argument.

a What is the length of the side of the large square in the figure? Write an expression for its area.

b Write another expression for the area of the large square by adding the areas of the four right triangles and the smaller central square.

c Equate your two expressions for the area of the large square, and deduce the Pythagorean theorem.

Calcworkshop

Angle Relationships Simply Explained w/ 11+ Step-by-Step Examples!

// Last Updated: January 21, 2020 - Watch Video //

In today’s lesson, you’re going to learn all about angle relationships and their measures.

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

We’ll walk through 11 step-by-step examples to ensure mastery.

Let’s dive in!

Angle Pair Relationship Names

In Geometry , there are five fundamental angle pair relationships:

• Complementary Angles
• Supplementary Angles
• Adjacent Angles
• Linear Pair
• Vertical Angles

1. Complementary Angles

Complementary angles are two positive angles whose sum is 90 degrees.

For example, complementary angles can be adjacent, as seen in with ∠ABD and ∠CBD in the image below. Or they can be two acute angles, like ∠MNP and ∠EFG, whose sum is equal to 90 degrees. Both of these graphics represent pairs of complementary angles.

Complementary Angles Example

2. Supplementary Angles

Supplementary angles are two positive angles whose sum is 180 degrees.

For example, supplementary angles may be adjacent, as seen in with ∠ABD and ∠CBD in the image below. Or they can be two angles, like ∠MNP and ∠KLR, whose sum is equal to 180 degrees. Both of these graphics represent pairs of supplementary angles.

Supplementary Angles Example

What is important to note is that both complementary and supplementary angles don’t always have to be adjacent angles.

3. Adjacent Angles

Adjacent angles are two angles in a plane that have a common vertex and a common side but no common interior points.

Angles 1 and 2 are adjacent angles because they share a common side.

Adjacent Angles Examples

And as Math is Fun so nicely points out, a straightforward way to remember Complementary and Supplementary measures is to think:

C is for Corner of a Right Angle (90 degrees) S is for Straight Angle (180 degrees)

Now it’s time to talk about my two favorite angle-pair relationships: Linear Pair and Vertical Angles.

4. Linear Pair

A linear pair is precisely what its name indicates. It is a pair of angles sitting on a line! In fact, a linear pair forms supplementary angles.

Because, we know that the measure of a straight angle is 180 degrees, so a linear pair of angles must also add up to 180 degrees.

∠ABD and ∠CBD form a linear pair and are also supplementary angles, where ∠1 + ∠2 = 180 degrees.

Linear Pair Example

5. Vertical Angles

Vertical angles are two nonadjacent angles formed by two intersecting lines or opposite rays.

Think of the letter X. These two intersecting lines form two sets of vertical angles (opposite angles). And more importantly, these vertical angles are congruent.

In the accompanying graphic, we see two intersecting lines, where ∠1 and ∠3 are vertical angles and are congruent. And ∠2 and ∠4 are vertical angles and are also congruent.

Vertical Angles Examples

Together we are going to use our knowledge of Angle Addition, Adjacent Angles, Complementary and Supplementary Angles, as well as Linear Pair and Vertical Angles to find the values of unknown measures.

Angle Relationships – Lesson & Examples (Video)

• Introduction to Angle Pair Relationships
• 00:00:15 – Overview of Complementary, Supplementary, Adjacent, and Vertical Angles and Linear Pair
• Exclusive Content for Member’s Only
• 00:06:29 – Use the diagram to solve for the unknown angle measures (Examples #1-8)
• 00:19:05 – Find the measure of each variable involving Linear Pair and Vertical Angles (Examples #9-12)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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Unit 8 Lesson 2 Homework (Applying Angle Relationships)

Chapter 2, Lesson 8: Proving Angle Relationships

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Angle relationships calculator.

Angles play a crucial role in geometry and trigonometry, and understanding their relationships is fundamental to solving various mathematical problems. Whether you’re a student struggling with geometry homework or a professional working with angles regularly, having a reliable tool to determine angle relationships can be immensely helpful.

1. Complementary Angles

Complementary angles are two angles that add up to 90 degrees. To find the complement of an angle, subtract its measure from 90 degrees. For example, if angle A measures 30 degrees, the complement of angle A would be 90 – 30 = 60 degrees.

2. Supplementary Angles

Supplementary angles are two angles that add up to 180 degrees. To find the supplement of an angle, subtract its measure from 180 degrees. For instance, if angle B measures 120 degrees, the supplement of angle B would be 180 – 120 = 60 degrees.

3. Adjacent Angles

Adjacent angles are two angles that share a common vertex and a common side, but do not overlap. In other words, they are side-by-side angles. Adjacent angles can be added together to find the total angle measure around a point, which is always 360 degrees.

4. Vertical Angles

Vertical angles are formed by two intersecting lines and are opposite each other. They have equal measures. Suppose angle C measures 40 degrees; then the vertical angle opposite angle C would also measure 40 degrees.

5. Corresponding Angles

Corresponding angles are formed when a transversal intersects two parallel lines. They lie on the same side of the transversal and in corresponding positions. Corresponding angles have equal measures. For example, if angle D measures 60 degrees, the corresponding angle on the other parallel line will also measure 60 degrees.

6. Conclusion

An angle relationships calculator can be immensely valuable in quickly determining the relationships between angles. By incorporating the formulas and rules mentioned above, you can easily find complementary, supplementary, adjacent, vertical, and corresponding angles. Whether you’re solving geometry problems or working on real-world applications, having a reliable calculator at your disposal saves time and minimizes the chance of errors.

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Line And Angle Relationships Calculator

Line And Angle Relationships Calculator In the world of mathematics understanding line and angle relationships is crucial. These relationships not only help us solve complex problems but also play a fundamental role in various disciplines such as engineering architecture and physics. However calcula – drawspaces.com

Angle Relationships With Parallel Lines Calculator

Angle Relationships With Parallel Lines Calculator Understanding angle relationships with parallel lines is an essential aspect of geometry. However calculating these angles manually can be time-consuming and prone to errors. Fortunately with the help of modern technology we now have access to angle – drawspaces.com

Triangle Calculator

This free triangle calculator computes the edges, angles, area, height, perimeter, median, as well as other values and a diagram of the resulting triangle. – www.calculator.net

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Complementary and Supplementary Angles Calculator

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Transversal Angle Calculator

Transversal Angle Calculator: Simplify Angle Measurements with Ease In the world of geometry understanding and calculating angles is essential for solving complex problems and real-world applications. One key aspect of this is transversal angles which play a crucial role in various scenarios. In thi – drawspaces.com

Transversal Of Parallel Lines Find Angle Measures Calculator

Transversal Of Parallel Lines Find Angle Measures Calculator Welcome to our comprehensive blog post on the topic of finding angle measures using a transversal of parallel lines calculator. Understanding the concept of transversals and parallel lines is crucial in geometry and this calculator will he – drawspaces.com

Corresponding Angle Calculator

Corresponding Angle Calculator: Simplify Angle Relationships with Ease Angles play a fundamental role in geometry engineering and various other disciplines. Understanding the relationships between angles is crucial for solving complex problems and gaining a deeper comprehension of spatial concepts. – drawspaces.com

Triangle Angle Calculator – CalcTree

Jul 6, 2023 … Need to calculate triangle angles? Look no further than CalcTree. Free-To-Use Triangle Angle Calculation tool. Try it now. – app.calctree.com

Supplementary Angles Calculator

Nov 20, 2023 … Supplementary angles relationships. Welcome to the supplementary angle calculator. You don’t need to rack your brain on how to find … – www.omnicalculator.com

Vertical Angles Relationship | Desmos

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Compressible Aerodynamics Calculator

Conical Shock Relations Perfect Gas, Gamma = , angles in degrees. INPUT: M1 = Cone angle, Wave angle, Mc. = Mc= – devenport.aoe.vt.edu

Solve for X Calculator – eMathHelp

The calculator will try to find the x (exact and numerical, real and complex) in the given equation. – www.emathhelp.net

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Basic geometry and measurement

Course: basic geometry and measurement   >   unit 11.

• Find measure of vertical angles

Finding missing angles

• Finding angle measures between intersecting lines
• Find measure of angles word problem
• Equation practice with complementary angles
• Equation practice with supplementary angles
• Equation practice with vertical angles
• Create equations to solve for missing angles
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

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Maneuvering the Middle

Student-Centered Math Lessons

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Angle Relationships Unit 8th Grade CCSS

Description, additional information.

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A 7 day CCSS-Aligned Angle Relationships Unit – including parallel lines and transversals, angle sum of triangles, exterior angles of triangles, and angle-angle criterion with your students.

Students will practice with both skill-based problems, real-world application questions, and error analysis to support higher level thinking skills.  You can reach your students and teach the standards without all of the prep and stress of creating materials!

Standards:   8.G.5;  Texas Teacher?  Grab the TEKS-Aligned Angle Relationships Unit.  Please don’t purchase both as there is overlapping content.

Learning Focus:

• use and apply angle relationships when parallel lines are cut by a transversal
• use angle-angle criterion to solve problems
• use facts about the angle sum and exterior angles of triangles

What is included in the 8th grade CCSS Angle Relationships Unit?

1. Unit Overviews

• Streamline planning with unit overviews that include essential questions, big ideas, vertical alignment, vocabulary, and common misconceptions.
• A pacing guide and tips for teaching each topic are included to help you be more efficient in your planning.

2. Student Handouts

• Student-friendly guided notes are scaffolded to support student learning.
• Available as a PDF and the student handouts/homework/study guides have been converted to Google Slides™ for your convenience.

3. Independent Practice

• Daily homework is aligned directly to the student handouts and is versatile for both in class or at home practice.

4. Assessments

• 1-2 quizzes, a unit study guide, and a unit test allow you to easily assess and meet the needs of your students.
• The Unit Test is available as an editable PPT, so that you can modify and adjust questions as needed.

5. Answer Keys

• All answer keys are included.

***Please download a preview to see sample pages and more information.***

How to use this resource:

• Use as a whole group, guided notes setting
• Use in a small group, math workshop setting
• Chunk each student handout to incorporate whole group instruction, small group practice, and independent practice.
• Incorporate our  Angle Relationships Activity Bundle  for hands-on activities as additional and engaging practice opportunities.

Time to Complete:

• Each student handout is designed for a single class period. However, feel free to review the problems and select specific ones to meet your student needs. There are multiple problems to practice the same concepts, so you can adjust as needed.

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• The unit test is editable with Microsoft PPT. The remainder of the file is a PDF and not editable.

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Angle Relationships Activity Bundle 8th Grade

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Angle Relationships Unit 8th Grade TEKS