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NYS COMMON CORE MATHEMATICS CURRICULUM 7ā¢6 Lesson 2
Lesson 2: Solving for Unknown Angles Using Equations
22
This work is derived from Eureka Math ā¢ and licensed by Great Minds. Ā©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 2: Solving for Unknown Angles Using Equations
Student Outcomes
Students solve for unknown angles in word problems and in diagrams involving complementary,
supplementary, vertical, and adjacent angles.
Classwork
Opening Exercise (5 minutes)
Opening Exercise
Two lines meet at a point. In a complete sentence, describe the
relevant angle relationships in the diagram. Find the values of š, š,
and š.
The two intersecting lines form two pairs of vertical angles;
š = šš, and šĀ° = šĀ°. Angles šĀ° and šĀ° are angles on a line and sum to
šššĀ°.
š = šš
š + šš = ššš
š + šš ā šš = ššš ā šš
š = ššš,
š = ššš
In the following examples and exercises, students set up and solve an equation for the
unknown angle based on the relevant angle relationships in the diagram. Model the
good habit of always stating the geometric reason when you use one. This is a
requirement in high school geometry.
Example 1 (4 minutes)
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 š and š.
The angle šĀ° is vertically opposite from and equal to the sum of the angles
with measurements ššĀ° and ššĀ°, or a sum of ššĀ°. Angles šĀ° and šĀ° are
angles on a line and sum to šššĀ°.
š = šš + šš
š = šš Vert. ā s
š + (šš) = ššš
š + šš ā šš = ššš ā šš
š = ššš
ā s on a line
25Ā°rĀ°
sĀ° tĀ°
16Ā°
pĀ°
rĀ°
28Ā°
Scaffolding:
Students may benefit from repeated practice drawing angle diagrams from verbal descriptions. For example, tell them āDraw a diagram of two supplementary angles, where one has a measure of 37Ā°.ā Students struggling to organize their solution to a problem may benefit from the five-part process of the Exit Ticket in Lesson 1, including writing an equation, explaining the connection between the equation and the situation, and assessing whether an answer is reasonable. This builds conceptual understanding.
MP.6
NYS COMMON CORE MATHEMATICS CURRICULUM 7ā¢6 Lesson 2
Lesson 2: Solving for Unknown Angles Using Equations
23
This work is derived from Eureka Math ā¢ and licensed by Great Minds. Ā©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Take the opportunity to distinguish the correct usage of supplementary versus angles on a line in this example. Remind
students that supplementary should be used in reference to two angles, whereas angles on a line can be used for two or
more angles.
Exercise 1 (4 minutes)
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 š.
The two šĀ° angles and the angle šššĀ° are angles on a line and
sum to šššĀ°.
šš + ššš = ššš
šš + ššš ā ššš = ššš ā ššš
šš = šš
š = šš
ā s on a line
Example 2 (4 minutes)
Encourage students to label diagrams as needed to facilitate their solutions. In this example, the label š¦Ā° is added to the
diagram to show the relationship of š§Ā° with 19Ā°. This addition allows for methodical progress toward the solution.
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 š.
Let šĀ° be the angle vertically opposite and equal in measurement to ššĀ°.
The angles šĀ° and šĀ° are complementary and sum to ššĀ°.
š + š = šš
š + šš = šš
š + šš ā šš = šš ā šš
š = šš
Complementary ā s
19Ā°
zĀ°šĖ
NYS COMMON CORE MATHEMATICS CURRICULUM 7ā¢6 Lesson 2
Lesson 2: Solving for Unknown Angles Using Equations
24
This work is derived from Eureka Math ā¢ and licensed by Great Minds. Ā©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exercise 2 (4 minutes)
Exercise 2
Three lines meet at a point; ā šØš¶š = šššĀ°. In a complete sentence, describe
the relevant angle relationships in the diagram. Set up and solve an equation to
determine the value of š.
ā š¬š¶š©, formed by adjacent angles ā š¬š¶šŖ and ā šŖš¶š©, is vertical to and equal in
measurement to ā šØš¶š.
The measurement of ā š¬š¶š© is šĀ° + ššĀ° (ā s add).
š + šš = ššš
š + šš ā šš = ššš ā šš
š = šš
Vert. ā s
Example 3 (4 minutes)
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 š.
The measurement of the angle formed by adjacent angles of ššĀ° and ššĀ° is
the sum of the adjacent angles. This angle is vertically opposite and equal in
measurement to the angle šĀ°.
Let šĀ° be the measure of the indicated angle.
š = ššš
š = (š)
š = ššš
ā s add
Vert. ā s
Exercise 3 (4 minutes)
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 š.
One possible response: Let šĀ° be the angle vertically opposite and equal
in measurement to ššĀ°. The angles šĀ° and šĀ° are adjacent angles, and
the angle they form together is equal to the sum of their measurements.
š = šš
š = šš + šš
š = ššš
Vert. ā s
ā s add
144Ā°
cĀ°
A B
C
D
E
F
O
46Ā°
vĀ°šĖ
26Ā°
tĀ°
šĖ
Ė
NYS COMMON CORE MATHEMATICS CURRICULUM 7ā¢6 Lesson 2
Lesson 2: Solving for Unknown Angles Using Equations
25
This work is derived from Eureka Math ā¢ and licensed by Great Minds. Ā©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Example 4 (4 minutes)
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 š. Is your answer reasonable? Explain how you know.
The angle šĀ° is vertically opposite from the angle formed by the right angle that
contains and shares a common side with an šĀ° angle.
š = šš ā š
š = šš
ā s add and vert. ā s
The answer is reasonable because the angle marked by šĀ° is close to appearing as
a right angle.
Exercise 4 (4 minutes)
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 š. Find the measurements of ā šØš¶š© and
ā š©š¶šŖ.
ā šØš¶šŖ is vertically opposite from the angle formed by adjacent
angles ššĀ° and ššĀ°.
šš + šš = šš + šš
šš = ššš
š = šš
ā s add and vert. ā s
ā šØš¶šŖ = š(šš)Ā° = ššĀ°
ā š©š¶šŖ = š(šš)Ā° = ššĀ°
xĀ°
8Ā°
25Ā°
2xĀ°
3xĀ°
A O
B
C
NYS COMMON CORE MATHEMATICS CURRICULUM 7ā¢6 Lesson 2
Lesson 2: Solving for Unknown Angles Using Equations
26
This work is derived from Eureka Math ā¢ and licensed by Great Minds. Ā©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exercise 5 (4 minutes)
Exercise 5
a. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation
to find the value of š. Find the measurements of ā šØš¶š© and ā š©š¶šŖ.
ā šØš¶š© and ā š©š¶šŖ are complementary and sum to ššĀ°.
šš + (šš + šš) = šš
šš + šš = šš
šš + šš ā šš = šš ā šš
šš = šš
š = šš
complementary ā s
ā šØš¶š© = š(šš)Ā° = ššĀ°
ā š©š¶šŖ = š(šš)Ā° + ššĀ° = ššĀ°
b. Katrina was solving the problem above and wrote the equation šš + šš = šš. Then, she rewrote this as
šš + šš = šš + šš. Why did she rewrite the equation in this way? How does this help her to find the value
of š?
She grouped the quantity on the right-hand side of the equation similarly to that of the left-hand side. This
way, it is clear that the quantity šš on the left-hand side must be equal to the quantity šš on the right-hand
side.
Closing (1 minute)
In every unknown angle problem, it is important to identify the angle relationship(s) correctly in order to set
up an equation that yields the unknown value.
Check your answer by substituting and/or measuring to be sure it is correct.
Exit Ticket (3 minutes)
5xĀ°
(2x+20)Ā°
AO
C
B
MP.7
Lesson Summary
To solve an unknown angle problem, identify the angle relationship(s) first to set up an equation that
will yield the unknown value.
Angles on a line and supplementary angles are not the same relationship. Supplementary angles are
two angles whose angle measures sum to šššĀ° whereas angles on a line are two or more adjacent
angles whose angle measures sum to šššĀ°.
NYS COMMON CORE MATHEMATICS CURRICULUM 7ā¢6 Lesson 2
Lesson 2: Solving for Unknown Angles Using Equations
27
This work is derived from Eureka Math ā¢ and licensed by Great Minds. Ā©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Name Date
Lesson 2: Solving for Unknown Angles Using Equations
Exit Ticket
Two lines meet at a point that is also the vertex of an angle. Set up and solve an equation to find the value of š„. Explain
why your answer is reasonable.
27Ā°
xĀ°
65Ā°
NYS COMMON CORE MATHEMATICS CURRICULUM 7ā¢6 Lesson 2
Lesson 2: Solving for Unknown Angles Using Equations
28
This work is derived from Eureka Math ā¢ and licensed by Great Minds. Ā©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exit Ticket Sample Solutions
Two lines meet at a point that is also the vertex of an angle. Set up and solve an equation to find the value of š. Explain
why your answer is reasonable.
šš + (šš ā šš) = š
š = ššš
OR
š + šš = šš
š + šš ā šš = šš ā šš
š = šš
šš + š = š
šš + (šš) = š
š = ššš
The answers seem reasonable because a rounded value of š as šš and a rounded value of its adjacent angle šš as šš
yields a sum of ššš, which is close to the calculated answer.
Problem Set Sample Solutions
Note: Arcs indicating unknown angles begin to be dropped from the diagrams. It is necessary for students to determine
the specific angle whose measure is being sought. Students should draw their own arcs.
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 š.
š + šš + šš = ššš
š + ššš = ššš
š + ššš ā ššš = ššš ā ššš
š = šš
ā s on a line
Scaffolded solutions:
a. Use the equation above.
b. The angle marked šĀ°, the right angle, and the angle with measurement ššĀ° are angles on a line, and their
measurements sum to šššĀ°.
c. Use the solution above. The answer seems reasonable because it looks like it has a measurement a little less
than a ššĀ° angle.
27Ā°
xĀ°
65Ā°šĖ
Scaffolding:
Students struggling to organize their solution may benefit from prompts such as the following: Write an equation to model this situation. Explain how your equation describes the situation. Solve and interpret the solution. Is it reasonable?
cĀ°17Ā°
NYS COMMON CORE MATHEMATICS CURRICULUM 7ā¢6 Lesson 2
Lesson 2: Solving for Unknown Angles Using Equations
29
This work is derived from Eureka Math ā¢ and licensed by Great Minds. Ā©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
2. 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 š.
Explain why your answer is reasonable.
š + šš = šš
š + šš ā šš = šš ā šš
š = šš
ā s add and vert. ā s
The answers seem reasonable because a rounded value of š as šš and a rounded
value of its adjacent angle šš as šš yields a sum of šš, which is close to the
rounded value of the measurement of the vertical angle.
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 š.
š + šš = ššš
š + šš ā šš = ššš ā šš
š = šš
ā s add and vert. ā s
4. Two lines meet at a point that is also the vertex of an angle. Set up and solve an equation to find the value of š.
(šš ā šš) + šš = š
š = šš
ā s add and vert. ā s
5. Three lines meet at a point. Set up and solve an equation to find the value of š.
š + ššš + šš = ššš
š + ššš = ššš
š + ššš ā ššš = ššš ā ššš
š = šš
ā s on a line and vert. ā s
33Ā°aĀ°
49Ā°
34Ā°
122Ā°rĀ°
68Ā°
mĀ°
24Ā°
NYS COMMON CORE MATHEMATICS CURRICULUM 7ā¢6 Lesson 2
Lesson 2: Solving for Unknown Angles Using Equations
30
This work is derived from Eureka Math ā¢ and licensed by Great Minds. Ā©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
51Ā° 43Ā°
xĀ°yĀ°
zĀ°
wĀ°
vĀ°
6. Three lines meet at a point that is also the endpoint of a ray. Set up and solve an equation to find the value of each
variable in the diagram.
š = šš ā šš
š = šš
Complementary ā s
š + šš + šš + šš = ššš
š + ššš = ššš
š + ššš ā ššš = ššš ā ššš
š = šš
ā s on a line
š = šš + šš
š = šš
Vert. ā s
š = šš
Vert. ā s
š = šš Vert. ā s
7. Set up and solve an equation to find the value of š. Find the measurement of ā šØš¶š© and of ā š©š¶šŖ.
(šš ā šš) + ššš = ššš
ššš ā šš = ššš
ššš ā šš + šš = ššš + šš
ššš = ššš
š = šš
Supplementary ā s
The measurement of ā šØš¶š©: š(šš)Ā° ā ššĀ° = ššĀ°
The measurement of ā š©š¶šŖ: šš(šš)Ā° = šššĀ°
Scaffolded solutions:
a. Use the equation above.
b. The marked angles are angles on a line, and their measurements sum to šššĀ°.
c. Once šš is substituted for š, then the measurement of ā šØš¶š© is ššĀ° and the measurement of ā š©š¶šŖ is šššĀ°.
These answers seem reasonable since ā šØš¶š© is acute and ā š©š¶šŖ is obtuse.
8. Set up and solve an equation to find the value of š. Find the measurement of ā šØš¶š© and of ā š©š¶šŖ.
š + š + šš = šš
šš + š = šš
šš + š ā š = šš ā š
šš = šš
š = ššš
š
Complementary ā s
The measurement of ā šØš¶š©: (šššš
) Ā° + šĀ° = šššš
Ā°
The measurement of ā š©š¶šŖ: š (šššš
) Ā° = šššš
Ā°
Scaffolding:
Students struggling to organize their solution may benefit from prompts such as the following:
Write an equation to
model this situation.
Explain how your equation
describes the situation.
Solve and interpret the
solution. Is it reasonable?
NYS COMMON CORE MATHEMATICS CURRICULUM 7ā¢6 Lesson 2
Lesson 2: Solving for Unknown Angles Using Equations
31
This work is derived from Eureka Math ā¢ and licensed by Great Minds. Ā©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
2xĀ°10xĀ°
(4x+5)Ā° (5x+22)Ā°
A O C
B
9. Set up and solve an equation to find the value of š. Find the measurement of ā šØš¶š© and of ā š©š¶šŖ.
šš + š + šš + šš = ššš
šš + šš = ššš
šš + šš ā šš = ššš ā šš
šš = ššš
š = šš
ā s on a line
The measurement of ā šØš¶š©: š(šš)Ā° + šĀ° = ššĀ°
The measurement of ā š©š¶šŖ: š(šš)Ā° + ššĀ° = šššĀ°
10. Write a verbal problem that models the following diagram. Then, solve for the two angles.
One possible response: Two angles are supplementary. The
measurement of one angle is five times the measurement of the
other. Find the measurements of both angles.
ššš + šš = ššš
ššš = ššš
š = šš
Supplementary ā s
The measurement of Angle 1: šš(šš)Ā° = šššĀ°
The measurement of Angle 2: š(šš)Ā° = ššĀ°