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7.1.4: Solving for Unknown Angles

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• Illustrative Mathematics
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Let's figure out some missing angles.

Exercise \(\PageIndex{1}\): True or False: Length Relationships

Here are some line segments.

Decide if each of these equations is true or false. Be prepared to explain your reasoning.

\(CD+BC=BD\)

\(AB+BD=CD+AD\)

\(AC-AB=AB\)

\(BD-CD=AC-AB\)

Exercise \(\PageIndex{2}\): Info Gap: ANgle Finding

Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner.

If your teacher gives you the problem card :

• Silently read your card and think about what information you need to be able to answer the question.
• Ask your partner for the specific information that you need.
• Explain how you are using the information to solve the problem. Continue to ask questions until you have enough information to solve the problem.
• Share the problem card and solve the problem independently.
• Read the data card and discuss your reasoning.

If your teacher gives you the data card :

• Silently read your card.
• Ask your partner “What specific information do you need?” and wait for them to ask for information. If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.
• Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions.
• Read the problem card and solve the problem independently.
• Share the data card and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Exercise \(\PageIndex{3}\): What's the Match?

Match each figure to an equation that represents what is seen in the figure. For each match, explain how you know they are a match.

• \(g+h=180\)
• \(2h+g=90\)
• \(g+h+48=180\)
• \(g+h+35=180\)

Are you ready for more?

• What is the angle between the hour and minute hands of a clock at 3:00?
• You might think that the angle between the hour and minute hands at 2:20 is 60 degrees, but it is not! The hour hand has moved beyond the 2. Calculate the angle between the clock hands at 2:20.
• Find a time where the hour and minute hand are 40 degrees apart. (Assume that the time has a whole number of minutes.) Is there just one answer?

We can write equations that represent relationships between angles.

• The first pair of angles are supplementary, so \(x+42=180\).
• The second pair of angles are vertical angles, so \(y=28\).
• Assuming the third pair of angles form a right angle, they are complementary, so \(z+64=90\).

Glossary Entries

Definition: Adjacent Angles

Adjacent angles share a side and a vertex.

In this diagram, angle \(ABC\) is adjacent to angle \(DBC\).

Definition: Complementary

Complementary angles have measures that add up to 90 degrees.

For example, a \(15^{\circ}\) angle and a \(75^{\circ}\) angle are complementary.

Definition: Right Angle

A right angle is half of a straight angle. It measures 90 degrees.

Definition: Straight Angle

A straight angle is an angle that forms a straight line. It measures 180 degrees.

Definition: Supplementary

Supplementary angles have measures that add up to 180 degrees.

For example, a \(15^{\circ}\) angle and a \(165^{\circ}\) angle are supplementary.

Definition: Vertical Angles

Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal.

For example, angles \(AEC\) and \(DEB\) are vertical angles. If angle \(AEC\) measures \(120^{\circ}\), then angle \(DEB\) must also measure \(120^{\circ}\).

Angles \(AED\) and \(BEC\) are another pair of vertical angles.

Exercise \(\PageIndex{4}\)

\(M\) is a point on line segment \(KL\). \(NM\) is a line segment. Select all the equations that represent the relationship between the measures of the angles in the figure.

• \(a+b=180\)
• \(180-a=b\)
• \(180=b-a\)

Exercise \(\PageIndex{5}\)

Which equation represents the relationship between the angles in the figure?

• \(88+b=90\)
• \(88+b=180\)
• \(2b+88=90\)
• \(2b+88=180\)

Exercise \(\PageIndex{6}\)

Segments \(AB\), \(EF\), and \(CD\) intersect at point \(C\), and angle \(ACD\) is a right angle. Find the value of \(g\).

Exercise \(\PageIndex{7}\)

Select all the expressions that are the result of decreasing \(x\) by 80%.

• \(\frac{20}{100}x\)
• \(x-\frac{80}{100}x\)
• \(\frac{100-20}{100}x\)
• \((1-0.8)x\)

(From Unit 6.2.6)

Exercise \(\PageIndex{8}\)

Andre is solving the equation \(4(x+\frac{3}{2})=7\). He says, “I can subtract \(\frac{3}{2}\) from each side to get \(4x=\frac{11}{2}\) and then divide by 4 to get \(x=\frac{11}{8}\).” Kiran says, “I think you made a mistake.”

• How can Kiran know for sure that Andre’s solution is incorrect?
• Describe Andre’s error and explain how to correct his work.

(From Unit 6.2.2)

Exercise \(\PageIndex{9}\)

Solve each equation.

\(\begin{array}{lllll}{\frac{1}{7}a+\frac{3}{4}=\frac{9}{8}}&{\qquad}&{\frac{2}{3}+\frac{1}{5}b=\frac{5}{6}}&{\qquad}&{\frac{3}{2}=\frac{4}{3}c+\frac{2}{3}}\\{0.3d+7.9=9.1}&{\qquad}&{11.03=8.78+0.02e}&{\qquad}&{\qquad}\end{array}\)

(From Unit 6.2.1)

Exercise \(\PageIndex{10}\)

A train travels at a constant speed for a long distance. Write the two constants of proportionality for the relationship between distance traveled and elapsed time. Explain what each of them means.

(From Unit 2.2.2)

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How to Solve Unknown Angles? (+FREE Worksheet!)

In this article, you will learn how to solve unknown angles in a few simple steps.

Step by step guide to Solve for unknown angles

To determine the measurement of an unknown angle, we must identify the angle relationship(s), and then model the relationship with an equation that will yield the unknown value.

If the sum of the measurements of two angles is \(90^{\circ}\), angles are complementary angles and one is the complement of the other. If two complementary angles are adjacent to each other (have a vertex and a common side), the other two non-common sides will form a right angle.

In geometry, two acute angles of a right triangle complement each other. Since the sum of the internal angles of the triangle must be \(180\) degrees, and the right angle of the triangle itself is \(90\) degrees, then the sum of the two remaining angles must be \(90\) degrees, and they will complement each other.

If the sum of the measurement of two angles is \(180^{\circ}\), angles are supplementary angles and one is the supplement of the other.

When two supplementary angles are adjacent to each other (have a vertex and a common side), the two non-common sides form a straight line or a straight angle. For example, the supplement of \(135^{\circ}\) angle is equal to \(135-180\). This means the supplement of the \(135^{\circ}\) angle is the \(45^{\circ}\) angle.

Supplementary angles can be separated and they do not have to be on a straight line. for example, adjacent angles of a parallelogram supplement each other in pairs.

Solving for unknown angles Example 1:

Find the measure of \(x\).

Solution: The two angles are a linear pair (from a straight line), So they must add to\(=180^{\circ}\)

Write an equation based on what you know: \(x+66=180\)

Solve the equation: \(x+66=180→x=180-66→x=114\)

The missing angle is \(x=114^{\circ}\)

Solving for unknown angles Example 2:

Find the missing angle.

Write an equation based on what you know: \(x+42=180\)

Solve the equation: \(x+42=180→x=180-42→x=138\)

The missing angle is \(x=138^{\circ}\)

Exercises for Solving for unknown angles

Find the measure of the missing angle..

• \(x=97^{\circ}\)
• \(x=49^{\circ}\)

by: Effortless Math Team about 3 years ago (category: Articles , Free Math Worksheets )

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Illustrative Mathematics Grade 7, Unit 7, Lesson 5: Using Equations to Solve for Unknown Angles

Learning Targets:

• I can write an equation to represent a relationship between angle measures and solve the equation to find unknown angle measures.

Related Pages Illustrative Math Grade 7

Lesson 5: Using Equations to Solve for Unknown Angles

Let’s figure out missing angles using equations.

Illustrative Math Unit 7.7, Lesson 5 (printable worksheets)

Lesson 5 Summary

Lesson 5.1 Is This Enough?

Tyler thinks that this figure has enough information to figure out the values of a and b. Do you agree? Explain your reasoning.

Lesson 5.2 What Does It Look Like?

Elena and Diego each wrote equations to represent these diagrams. For each diagram, decide which equation you agree with, and solve it. You can assume that angles that look like right angles are indeed right angles.

• Elena: x = 35 Diego: x + 35 = 180
• Elena: 5 + w + 41 = 180 Diego: 35 + w = 180
• Elena: w + 35 = 90 Diego: 2w + 35 = 90
• Elena: 2w + 35 = 90 Diego: w + 35 = 90
• Elena: x + 148 + 180 Diego: x + 90 =148

Lesson 5.3 Calculate the Measure

Find the unknown angle measures. Show your thinking. Organize it so it can be followed by others.

Are you ready for more?

The diagram contains three squares. Three additional segments have been drawn that connect corners of the squares. We want to find the exact value of a + b + c.

• Use a protractor to measure the three angles. Use your measurements to conjecture about the value of a + b + c.

Lesson 5 Practice Problems

• Segments AB, BC, and EC intersect at point C. Angle DCE measures 148°. Find the value of x.
• Line l is perpendicular to line m. Find the value of x and w.
• If you knew that two angles were complementary and were given the measure of one of those angles, would you be able to find the measure of the other angle? Explain your reasoning.
• For each inequality, decide whether the solution is represented by x < 4.5 or x > 4.5.
• A runner ran 2/3 of a 5 kilometer race in 21 minutes. They ran the entire race at a constant speed. a. How long did it take to run the entire race? b. How many minutes did it take to run 1 kilometer?
• Jada, Elena, and Lin walked a total of 37 miles last week. Jada walked 4 more miles than Elena, and Lin walked 2 more miles than Jada. The diagram represents this situation: Find the number of miles that they each walked. Explain or show your reasoning.
• Select all the expressions that are equivalent to -36x + 54y - 90

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Angle relationships and unknown angle problems

This lesson first explains supplementary and vertical angles. Then we look at a variety of "angle puzzles", or problems with an unknown angle, and solve those using these basic principles. I also write an equation for each problem, to help students learn to both write and solve equations, since this lesson belongs to my pre-algebra course .

Angles in a triangle — video lesson

Math Mammoth Geometry 3 — a self-teaching worktext with explanations & exercises (grades 4-5)

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RadfordMathematics.com

Online mathematics book, search radford mathematics, unknown angles in right angle triangles - soh cah toa, (trigonometric ratios).

In this section we learn how to use SOH CAH TOA to find unknown angles in right angle triangles .

What You'll find here:

• We start this section by watching a tutorial
• We then write a three step method for finding angles, that will always work (do make a note of it).
• Practice exercises , that can be downloaded as a .pdf worksheet .

SOH CAH TOA (Reminder)

SOH CAH TOA are three short words that help us remember the trigonometric ratios , sine , cosine and tangent .

Given one of the two acute angles inside a right angle triangle , we'll call that angle a , here's what SOH CAH TOA tells us:

• SOH: the S ine of angle a equals to its O pposite side length divided by the H ypotenuse; that's: \[sin(a) = \frac{O}{H}\]
• CAH: the C osine of angle a equals to its A adjacent side length divided by the H ypotenuse; that's: \[cos(a) = \frac{A}{H}\]
• TOA: the T angent of angle a equals to its O pposite side length divided by the A djacent side length; that's: \[tan(a) = \frac{O}{A}\]

Tutorial: Unknown Angles Using SOH CAH TOA

In the following tutorial we learn how to find unknown angles in right angle triangles , using the trigonometric ratios and SOH CAH TOA .

Given a right angle triangle, the method for finding an unknown angle \(a\) , can be summarized in three steps :

• Step 1: Label the side lengths, relative to the angle we're after, using "A", "O" and "H".
• Step 2: Using the labels, made in step 1 , look for the only one of the words "SOH", "CAH", or "TOA" that contains both of the letters "O" and "H", or "A" and "H", or "O" and "A". Write the corresponding trigonometric ratio for the unknown angle; \(sin(a) = \frac{O}{H}\), \(cos(a) = \frac{A}{H}\), or \(tan(a) = \frac{O}{A}\).
• Step 3: replace the letters, "O" and "H", or "A" and "H", or "O" and "A", by their actual values and find the angle using the correct inverse trigonometric function .

In each of the following right angle triangles, find the unknown side length marked x :

Solution Without Working

• \(x = 36.9^{\circ}\)
• \(x = 26.6^{\circ}\)
• \(x = 69.4^{\circ}\)
• \(x = 60.0^{\circ}\)
• \(x = 46.6^{\circ}\)
• \(x = 34.2^{\circ}\)
• \(x = 63.5^{\circ}\)
• \(x = 32.0^{\circ}\)

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Addition Of Adjacent Angle Measures

Description.

Objective:  Use the addition of adjacent angle measures to solve problems using a symbol for the unknown angle measure.

In Lesson 10, students use what they know about the additive nature of angle measure to reason about the relationships between pairs of adjacent angles. Students discover that the measures of two angles on a straight line add up to 180° (supplementary angles) and that the measures of two angles meeting to form a right angle add up to 90° (complementary angles). In both Lessons 10 and 11, students write addition and subtraction equations to solve unknown angle problems. Students solve these problems using a variety of pictorial and numerical strategies, combined with the use of a protractor to verify answers (4.MD.7).

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• Grade 4 Mathematics Module 4, Topic C, Lesson 10

Prerequisites

• CCSS Standard:

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Unit 11: Angle relationships

Vertical, complementary, and supplementary angles.

• Complementary & supplementary angles (Opens a modal)
• Vertical angles (Opens a modal)
• Complementary and supplementary angles review (Opens a modal)
• Identifying supplementary, complementary, and vertical angles Get 5 of 7 questions to level up!
• Complementary and supplementary angles (visual) Get 3 of 4 questions to level up!
• Vertical angles Get 3 of 4 questions to level up!

Missing angles problems

• Find measure of vertical angles (Opens a modal)
• Find measure of angles word problem (Opens a modal)
• Equation practice with complementary angles (Opens a modal)
• Equation practice with supplementary angles (Opens a modal)
• Equation practice with vertical angles (Opens a modal)
• Finding missing angles Get 5 of 7 questions to level up!
• Finding angle measures between intersecting lines Get 3 of 4 questions to level up!
• Create equations to solve for missing angles Get 5 of 7 questions to level up!

Parallel lines and transversals

• Angles, parallel lines, & transversals (Opens a modal)
• Parallel & perpendicular lines (Opens a modal)
• Missing angles with a transversal (Opens a modal)
• Measures of angles formed by a transversal (Opens a modal)
• Angle relationships with parallel lines Get 5 of 7 questions to level up!

Triangle angles

• Angles in a triangle sum to 180° proof (Opens a modal)
• Isosceles & equilateral triangles problems (Opens a modal)
• Triangle exterior angle example (Opens a modal)
• Worked example: Triangle angles (intersecting lines) (Opens a modal)
• Worked example: Triangle angles (diagram) (Opens a modal)
• Triangle angle challenge problem (Opens a modal)
• Triangle angle challenge problem 2 (Opens a modal)
• Triangle angles review (Opens a modal)
• Find angles in triangles Get 5 of 7 questions to level up!
• Find angles in isosceles triangles Get 3 of 4 questions to level up!
• Finding angle measures using triangles Get 5 of 7 questions to level up!