Lesson 10: Angle Problems and Solving Equations · Lesson 10 . 7 • 3 . Lesson 10 : Angle Problems and Solving Equations . Exercise 2 . In a complete sentence, describe the angle

7•3 Lesson 10

Lesson 10: Angle Problems and Solving Equations


Classwork Angle Facts and Definitions

Name of Angle Relationship

Angle Fact Diagram

Adjacent Angles

Vertical Angles (vert. ∠s)

Angles on a Line (∠s on a line)

Angles at a Point (∠s at a point)

A STORY OF RATIOS

This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka- math.orgG7-M3-SE-1.3.0-07.2015

S.1

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US




7•3 Lesson 10


Opening Exercise

Use the diagram to complete the chart.

Name the angles that are …

Vertical

Adjacent

Angles on a line

Angles at a point

Example 1

Estimate the measurement of 𝑥𝑥. ________

In a complete sentence, describe the angle relationship in the diagram.

Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥. Then, find the measures of ∠𝐵𝐵𝐵𝐵𝐵𝐵 and confirm your answers by measuring the angle with a protractor.

A STORY OF RATIOS


S.2





7•3 Lesson 10


Exercise 1


Find the measurements of ∠𝐵𝐵𝐵𝐵𝐵𝐵 and ∠𝐷𝐷𝐵𝐵𝐷𝐷.

Example 2


Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥 and 𝑦𝑦. Find the measurements of ∠𝐿𝐿𝐷𝐷𝐵𝐵 and ∠𝐾𝐾𝐷𝐷𝐵𝐵.

A STORY OF RATIOS


S.3





7•3 Lesson 10


Exercise 2

In a complete sentence, describe the angle relationships in the diagram.

Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥.

Example 3


Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥. Find the measurement of ∠𝐷𝐷𝐾𝐾𝐸𝐸 and confirm your answers by measuring the angle with a protractor.

A STORY OF RATIOS


S.4





7•3 Lesson 10


Exercise 3


Find the measurement of ∠𝐺𝐺𝐵𝐵𝐺𝐺.

Example 4

The following two lines intersect. The ratio of the measurements of the obtuse angle to the acute angle in any adjacent angle pair in this figure is 2 ∶ 1. In a complete sentence, describe the angle relationships in the diagram.

Label the diagram with expressions that describe this relationship. Write an equation that models the angle relationship and solve for 𝑥𝑥. Find the measurements of the acute and obtuse angles.

A STORY OF RATIOS


S.5





7•3 Lesson 10


Exercise 4

The ratio of 𝑚𝑚∠𝐺𝐺𝐸𝐸𝐺𝐺 to 𝑚𝑚∠𝐷𝐷𝐸𝐸𝐺𝐺 is 2 ∶ 3. In a complete sentence, describe the angle relationships in the diagram.

Find the measures of ∠𝐺𝐺𝐸𝐸𝐺𝐺 and ∠𝐷𝐷𝐸𝐸𝐺𝐺.

Relevant Vocabulary

ADJACENT ANGLES: Two angles ∠𝐵𝐵𝐵𝐵𝐵𝐵 and ∠𝐵𝐵𝐵𝐵𝐷𝐷 with a common side 𝐵𝐵𝐵𝐵��⃗ are adjacent angles if 𝐵𝐵 belongs to the interior of ∠𝐵𝐵𝐵𝐵𝐷𝐷.

VERTICAL ANGLES: Two angles are vertical angles (or vertically opposite angles) if their sides form two pairs of opposite rays.

ANGLES ON A LINE: The sum of the measures of adjacent angles on a line is 180°.

ANGLES AT A POINT: The sum of the measures of adjacent angles at a point is 360°.

A STORY OF RATIOS


S.6





7•3 Lesson 10


Problem Set

For each question, use angle relationships to write an equation in order to solve for each variable. Determine the indicated angles. You can check your answers by measuring each angle with a protractor.

1. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurementof ∠𝐷𝐷𝐵𝐵𝐷𝐷.

2. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurementof ∠𝑄𝑄𝑄𝑄𝑄𝑄.

3. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurementsof ∠𝐵𝐵𝑄𝑄𝐷𝐷 and ∠𝐷𝐷𝑄𝑄𝐸𝐸.

4. In a complete sentence, describe the relevant angle relationships in thefollowing diagram. Find the measure of 𝑥𝑥.

A STORY OF RATIOS


S.7





7•3 Lesson 10


5. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures of 𝑥𝑥and 𝑦𝑦.

6. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures of 𝑥𝑥and 𝑦𝑦.

7. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measuresof ∠𝐵𝐵𝐵𝐵𝐷𝐷 and ∠𝐷𝐷𝐵𝐵𝐷𝐷.

A STORY OF RATIOS


S.8





7•3 Lesson 10


8. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of∠𝐵𝐵𝑄𝑄𝐺𝐺.

9. The ratio of the measures of a pair of adjacent angles on a line is 4 ∶ 5.

a. Find the measures of the two angles.

b. Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle.

10. The ratio of the measures of three adjacent angles on a line is 3 ∶ 4 ∶ 5.

a. Find the measures of the three angles.

b. Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle.

A STORY OF RATIOS


S.9





7•3 Lesson 11



Classwork

Opening Exercise

a. In a complete sentence, describe the angle relationship in the diagram. Write an equation for the anglerelationship shown in the figure and solve for 𝑥𝑥. Confirm your answers by measuring the angle with aprotractor.

b. 𝐶𝐶𝐶𝐶�⃖��⃗ and 𝐸𝐸𝐸𝐸�⃖��⃗ are intersecting lines. In a complete sentence, describe the angle relationship in the diagram.Write an equation for the angle relationship shown in the figure and solve for 𝑦𝑦. Confirm your answers bymeasuring the angle with a protractor.

A STORY OF RATIOS


S.10





7•3 Lesson 11


c. In a complete sentence, describe the angle relationship in the diagram. Write an equation for the anglerelationship shown in the figure and solve for 𝑏𝑏. Confirm your answers by measuring the angle with aprotractor.

d. The following figure shows three lines intersecting at a point. In a complete sentence, describe the anglerelationship in the diagram. Write an equation for the angle relationship shown in the figure and solve for 𝑧𝑧.Confirm your answers by measuring the angle with a protractor.

e. Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥. In a complete sentence,describe the angle relationship in the diagram. Find the measurements of ∠𝐸𝐸𝐸𝐸𝐸𝐸 and ∠𝐶𝐶𝐸𝐸𝐶𝐶. Confirm youranswers by measuring the angle with a protractor.

A STORY OF RATIOS


S.11





7•3 Lesson 11


Example 1

The following figure shows three lines intersecting at a point. In a complete sentence, describe the angle relationship in the diagram. Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥. Confirm your answers by measuring the angle with a protractor.

Exercise 1

The following figure shows four lines intersecting at a point. In a complete sentence, describe the angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥 and 𝑦𝑦. Confirm your answers by measuring the angle with a protractor.

Example 2

In a complete sentence, describe the angle relationships in the diagram. You may label the diagram to help describe the angle relationships. Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥. Confirm your answers by measuring the angle with a protractor.

A STORY OF RATIOS


S.12





7•3 Lesson 11


Exercise 2

In a complete sentence, describe the angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥 and 𝑦𝑦. Confirm your answers by measuring the angles with a protractor.

Example 3

In a complete sentence, describe the angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥. Find the measures of ∠𝐽𝐽𝐶𝐶𝐽𝐽 and ∠𝐺𝐺𝐶𝐶𝐸𝐸. Confirm your answers by measuring the angle with a protractor.

Exercise 3

In a complete sentence, describe the angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for 𝑥𝑥. Find the measures of ∠𝐽𝐽𝐽𝐽𝐺𝐺. Confirm your answer by measuring the angle with a protractor.

A STORY OF RATIOS


S.13





7•3 Lesson 11


Example 4

In the accompanying diagram, the measure of ∠𝐶𝐶𝐸𝐸𝐸𝐸 is four times the measure of ∠𝐸𝐸𝐸𝐸𝐺𝐺.

a. Label ∠𝐶𝐶𝐸𝐸𝐸𝐸 as 𝑦𝑦° and ∠𝐸𝐸𝐸𝐸𝐺𝐺 as 𝑥𝑥°. Write an equation thatdescribes the relationship between ∠𝐶𝐶𝐸𝐸𝐸𝐸 and ∠𝐸𝐸𝐸𝐸𝐺𝐺.

b. Find the value of 𝑥𝑥.

c. Find the measures of ∠𝐸𝐸𝐸𝐸𝐺𝐺, ∠𝐶𝐶𝐸𝐸𝐶𝐶, ∠𝐶𝐶𝐸𝐸𝐸𝐸, ∠𝐺𝐺𝐸𝐸𝐸𝐸, and ∠𝐶𝐶𝐸𝐸𝐸𝐸.

d. What is the measure of ∠𝐶𝐶𝐸𝐸𝐺𝐺? Identify the angle relationship used to get your answer.

A STORY OF RATIOS


S.14





7•3 Lesson 11


Problem Set

In a complete sentence, describe the angle relationships in each diagram. Write an equation for the angle relationship(s) shown in the figure, and solve for the indicated unknown angle. You can check your answers by measuring each angle with a protractor.

1. Find the measures of ∠𝐸𝐸𝐶𝐶𝐸𝐸, ∠𝐶𝐶𝐶𝐶𝐸𝐸, and ∠𝐶𝐶𝐶𝐶𝐶𝐶.

2. Find the measure of 𝑎𝑎.

3. Find the measures of 𝑥𝑥 and 𝑦𝑦.

A STORY OF RATIOS


S.15





7•3 Lesson 11


4. Find the measure of ∠𝐽𝐽𝐶𝐶𝐽𝐽.

5. Find the measures of ∠𝐽𝐽𝐶𝐶𝐸𝐸 and ∠𝐶𝐶𝐶𝐶𝐸𝐸.

6. The measure of ∠𝑆𝑆𝐸𝐸𝑆𝑆 is 𝑏𝑏°. The measure of ∠𝑆𝑆𝐸𝐸𝑇𝑇 is five more than two times ∠𝑆𝑆𝐸𝐸𝑆𝑆. The measure of ∠𝑄𝑄𝐸𝐸𝑆𝑆 is twelve less than eight times the measure of ∠𝑆𝑆𝐸𝐸𝑆𝑆. Find the measures of ∠𝑆𝑆𝐸𝐸𝑆𝑆, ∠𝑆𝑆𝐸𝐸𝑇𝑇, and ∠𝑄𝑄𝐸𝐸𝑆𝑆.

A STORY OF RATIOS


S.16





7•3 Lesson 11


7. Find the measures of ∠𝐽𝐽𝑄𝑄𝐸𝐸 and ∠𝐶𝐶𝑄𝑄𝐺𝐺.

8. The measures of three angles at a point are in the ratio of 2 ∶ 3 ∶ 5. Find the measures of the angles.

9. The sum of the measures of two adjacent angles is 72°. The ratio of the smaller angle to the larger angle is 1 ∶ 3.Find the measures of each angle.

10. Find the measures of ∠𝐶𝐶𝑄𝑄𝐶𝐶 and ∠𝐸𝐸𝑄𝑄𝐸𝐸.

A STORY OF RATIOS


S.17





7•6 Lesson 1

Lesson 1: Complementary and Supplementary Angles


Classwork

Opening Exercise

As we begin our study of unknown angles, let us review key definitions.

Term Definition

Two angles, ∠𝐴𝐴𝐴𝐴𝐴𝐴 and ∠𝐴𝐴𝐴𝐴𝐶𝐶, with a common side 𝐴𝐴𝐴𝐴��⃑ , are angles if 𝐴𝐴 is in the interior of ∠𝐴𝐴𝐴𝐴𝐶𝐶.

When two lines intersect, any two non-adjacent angles formed by those lines are called angles, or angles.

Two lines are if they intersect in one point, and any of the angles formed by the intersection of the lines is 90°. Two segments or rays are if the lines containing them are lines.

Complete the missing information in the table below. In the Statement column, use the illustration to write an equation that demonstrates the angle relationship; use all forms of angle notation in your equations.

Angle Relationship

Abbreviation Statement Illustration

Adjacent Angles

The measurements of adjacent angles add.

Vertical Angles

Vertical angles have equal measures.

A STORY OF RATIOS

S.18

©2015 G re at Min ds. eureka-math.orgG7-M6-SE-1.3.0-10.2015




7•6 Lesson 1


Angles on a Line

If the vertex of a ray lies on a line but the ray is not contained in that line, then the sum of measurements of the two angles formed is 180°.

Angles at a Point

Suppose three or more rays with the same vertex separate the plane into angles with disjointed interiors. Then, the sum of all the measurements of the angles is 360°.

Angle Relationship

Definition Diagram

Complementary Angles

Supplementary Angles

A STORY OF RATIOS

S.19





7•6 Lesson 1


Exercise 1

1. In a complete sentence, describe the relevant anglerelationships in the diagram. Write an equation for the anglerelationship shown in the figure and solve for 𝑥𝑥. Confirm youranswers by measuring the angle with a protractor.

Example 1

The measures of two supplementary angles are in the ratio of 2: 3. Find the measurements of the two angles.

Exercises 2–4

2. In a pair of complementary angles, the measurement of the larger angle is three times that of the smaller angle.Find the measurements of the two angles.

A STORY OF RATIOS

S.20





7•6 Lesson 1


3. The measure of a supplement of an angle is 6° more than twice the measure of the angle. Find the measurement ofthe two angles.

4. The measure of a complement of an angle is 32° more than three times the angle. Find the measurement of thetwo angles.

Example 2

Two lines meet at a point that is also the vertex of an angle. Set up and solve an appropriate equation for 𝑥𝑥 and 𝑦𝑦.

A STORY OF RATIOS

S.21





7•6 Lesson 1


Problem Set

1. Two lines meet at a point that is also the endpoint of a ray.Set up and solve the appropriate equations to determine 𝑥𝑥and 𝑦𝑦.

2. Two lines meet at a point that is also the vertex of an angle. Set up andsolve the appropriate equations to determine 𝑥𝑥 and 𝑦𝑦.

Lesson Summary

Supplementary angles are two angles whose measurements sum to 180°.

Complementary angles are two angles whose measurements sum to 90°.

Once an angle relationship is identified, the relationship can be modeled with an equation that will find anunknown value. The unknown value may be used to find the measure of the unknown angle.

A STORY OF RATIOS

S.22





7•6 Lesson 1


3. Two lines meet at a point that is also the vertex of an angle. Set upand solve an appropriate equation for 𝑥𝑥 and 𝑦𝑦.

4. Set up and solve the appropriate equations for 𝑠𝑠 and 𝑡𝑡.

5. Two lines meet at a point that is also the endpoint of two rays. Set up and solve the appropriate equations for 𝑚𝑚and 𝑛𝑛.

6. The supplement of the measurement of an angle is 16° less than three times the angle. Find the measurement ofthe angle and its supplement.

7. The measurement of the complement of an angle exceeds the measure of the angle by 25%. Find themeasurement of the angle and its complement.

A STORY OF RATIOS

S.23





7•6 Lesson 1


8. The ratio of the measurement of an angle to its complement is 1: 2. Find the measurement of the angle and itscomplement.

9. The ratio of the measurement of an angle to its supplement is 3: 5. Find the measurement of the angle and itssupplement.

10. Let 𝑥𝑥 represent the measurement of an acute angle in degrees. The ratio of the complement of 𝑥𝑥 to the supplementof 𝑥𝑥 is 2: 5. Guess and check to determine the value of 𝑥𝑥. Explain why your answer is correct.

A STORY OF RATIOS

S.24





7•6 Lesson 2

Lesson 2: Solving for Unknown Angles Using Equations


Classwork

Opening Exercise

Two lines meet at a point. In a complete sentence, describe the relevant angle relationships in the diagram. Find the values of 𝑟𝑟, 𝑠𝑠, and 𝑡𝑡.

Example 1

Two lines meet at a point that is also the endpoint of a ray. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑝𝑝 and 𝑟𝑟.

A STORY OF RATIOS

S.25





7•6 Lesson 2


Exercise 1

Three lines meet at a point. In a complete sentence, describe the relevant angle relationship in the diagram. Set up and solve an equation to find the value of 𝑎𝑎.

Example 2

Three lines meet at a point. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑧𝑧.

A STORY OF RATIOS

S.26





7•6 Lesson 2


Exercise 2

Three lines meet at a point; ∠𝐴𝐴𝐴𝐴𝐴𝐴 = 144°. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to determine the value of 𝑐𝑐.

Example 3

Two lines meet at a point that is also the endpoint of a ray. The ray is perpendicular to one of the lines as shown. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑡𝑡.

A STORY OF RATIOS

S.27





7•6 Lesson 2


Exercise 3

Two lines meet at a point that is also the endpoint of a ray. The ray is perpendicular to one of the lines as shown. In a complete sentence, describe the relevant angle relationships in the diagram. You may add labels to the diagram to help with your description of the angle relationship. Set up and solve an equation to find the value of 𝑣𝑣.

Example 4

Three lines meet at a point. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥𝑥. Is your answer reasonable? Explain how you know.

A STORY OF RATIOS

S.28





7•6 Lesson 2


Exercise 4

Two lines meet at a point that is also the endpoint of two rays. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥𝑥. Find the measurements of ∠𝐴𝐴𝐴𝐴𝐴𝐴 and ∠𝐴𝐴𝐴𝐴𝐵𝐵.

Exercise 5

a. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equationto find the value of 𝑥𝑥. Find the measurements of ∠𝐴𝐴𝐴𝐴𝐴𝐴 and∠𝐴𝐴𝐴𝐴𝐵𝐵.

A STORY OF RATIOS

S.29





7•6 Lesson 2


b. Katrina was solving the problem above and wrote the equation 7𝑥𝑥 + 20 = 90. Then, she rewrote this as7𝑥𝑥 + 20 = 70 + 20. Why did she rewrite the equation in this way? How does this help her to find thevalue of 𝑥𝑥?

A STORY OF RATIOS

S.30





7•6 Lesson 2


Problem Set

1. Two lines meet at a point that is also the endpoint of a ray.Set up and solve an equation to find the value of 𝑐𝑐.

2. Two lines meet at a point that is also the endpoint of a ray. Set up and solve an equationto find the value of 𝑎𝑎. Explain why your answer is reasonable.

3. Two lines meet at a point that is also the endpoint of a ray. Set up and solve an equation to find the value of 𝑤𝑤.

Lesson Summary

To solve an unknown angle problem, identify the angle relationship(s) first to set up an equation that willyield the unknown value.

Angles on a line and supplementary angles are not the same relationship. Supplementary angles are twoangles whose angle measures sum to 180° whereas angles on a line are two or more adjacent angleswhose angle measures sum to 180°.

A STORY OF RATIOS

S.31





7•6 Lesson 2


4. Two lines meet at a point that is also the vertex of an angle. Set up and solve anequation to find the value of 𝑚𝑚.

5. Three lines meet at a point. Set up and solve an equation to find the value of 𝑟𝑟.

6. Three lines meet at a point that is also the endpoint of a ray. Set up and solvean equation to find the value of each variable in the diagram.

7. Set up and solve an equation to find the value of 𝑥𝑥. Find the measurement of ∠𝐴𝐴𝐴𝐴𝐴𝐴 and of ∠𝐴𝐴𝐴𝐴𝐵𝐵.

A STORY OF RATIOS

S.32





7•6 Lesson 2




10. Write a verbal problem that models the following diagram. Then, solve for the two angles.

A STORY OF RATIOS

S.33





Documents

Lesson 10: Angle Problems and Solving Equations · Lesson 10 . 7 • 3 . Lesson 10 : Angle Problems and Solving Equations . Exercise 2 . In a complete sentence, describe the angle