7•3
Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 10: Angle Problems and Solving Equations
Classwork
Angle Facts and Definitions
Name of Angle
Relationship
Angle Fact
Diagram
B
Adjacent Angles
C
a°
b°
A
D
D
Vertical Angles
(vert. ∠s)
G
a°
b°
C
F
E
C
D
Angles on a Line
(∠s on a line)
b°
a°
A
B
c°
E
B
Angles at a Point
(∠s at a point)
a°
b°
A
D
c°
C
Lesson 10:
Date:
Angle Problems and Solving Equations
2/9/16
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
7•3
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 𝑥.
________
In a complete sentence, describe the angle relationship in the
diagram.
C
x°
B
132°
A
D
Write an equation for the angle relationship shown in the figure and solve for 𝑥. Then find the measures of ∠𝐵𝐴𝐶 and
confirm your answers by measuring the angle with a protractor.
Lesson 10:
Date:
Angle Problems and Solving Equations
2/9/16
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7•3
Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
Exercise 1
In a complete sentence, describe the angle relationship in the diagram.
C
D
3x°
2x°
A
B
E
Find the measurements of ∠𝐵𝐴𝐶 and ∠𝐷𝐴𝐸.
Example 2
L
In a complete sentence, describe the angle relationship in the diagram.
144°
A
x°
E
B
y°
K
Write an equation for the angle relationship shown in the figure and solve for 𝑥 and 𝑦. Find the measurements of ∠𝐿𝐸𝐵
and ∠𝐾𝐸𝐵.
Lesson 10:
Date:
Angle Problems and Solving Equations
2/9/16
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
7•3
Exercise 2
In a complete sentence, describe the angle relationships in the diagram.
N
16° M
J
3x°
E
85°
Write an equation for the angle relationship shown in the figure and solve for 𝑥.
L
K
Example 3
In a complete sentence, describe the angle relationships in the diagram.
E
x°
G
K
135°
Write an equation for the angle relationship shown in the figure and solve for
𝑥. Find the measurement of ∠𝐸𝐾𝐹 and confirm your answers by measuring
the angle with a protractor.
Lesson 10:
Date:
F
Angle Problems and Solving Equations
2/9/16
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
7•3
Exercise 3
In a complete sentence, describe the angle relationships in the diagram.
F
E
103°
A
59°
G
(x+1)°
167°
Find the measurement of ∠𝐺𝐴𝐻.
H
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 𝑥. Find the measurements of the acute and obtuse angles.
Lesson 10:
Date:
Angle Problems and Solving Equations
2/9/16
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
7•3
Exercise 4
The ratio of ∠𝐺𝐹𝐻 to ∠𝐸𝐹𝐻 is 2 ∶ 3. In a complete sentence, describe the angle relationships in the diagram.
E
H
3x°
2x°
G
F
Find the measures of ∠𝐺𝐹𝐻 and ∠𝐸𝐹𝐻.
Relevant Vocabulary
ADJACENT ANGLES: Two angles ∠𝐵𝐴𝐶 and ∠𝐶𝐴𝐷 with a common side ⃗⃗⃗⃗⃗
𝐴𝐶 are adjacent angles if 𝐶 belongs to the interior
of ∠𝐵𝐴𝐷.
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°.
Lesson 10:
Date:
Angle Problems and Solving Equations
2/9/16
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
7•3
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 measurement
of ∠𝐷𝐴𝐸.
D
E
x°
65°
A
C
F
2.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurement
of ∠𝑄𝑃𝑅.
3.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurements
of ∠𝐶𝑄𝐷 and ∠𝐸𝑄𝐹.
E
D
2x°
C
10°
B
4.
103°
3x°
Q
12°
F
G
In a complete sentence, describe the relevant angle relationships in
the following diagram. Find the measure of 𝑥.
x°
71°
x°
71°
71°
x°
71°
Lesson 10:
Date:
x°
Angle Problems and Solving Equations
2/9/16
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
7•3
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures of 𝑥
and 𝑦.
B
D
25°
K
E
F
x°
y°
C
6.
A
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures of 𝑥
and 𝑦.
E
F
x°
G
x°
C
24°
A
D
y°
K
y° y°
J
H
F
7.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures
of ∠𝐶𝐴𝐷 and ∠𝐷𝐴𝐸.
Lesson 10:
Date:
Angle Problems and Solving Equations
2/9/16
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Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
8.
7•3
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of
∠𝐶𝑄𝐺.
G
3x°
F
56°
Q
155°
C
D
E
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.
c.
Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle.
Lesson 10:
Date:
Angle Problems and Solving Equations
2/9/16
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.67
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.