NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
150
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Lesson 10: Angle Problems and Solving Equations
Student Outcomes
Students use vertical angles, adjacent angles, angles on a line, and angles at a point in a multistep problem to
write and solve simple equations for an unknown angle in a figure.
Lesson Notes
In Lessons 10 and 11, students apply their understanding of equations to unknown angle problems. The geometry topic
is a natural context within which algebraic skills are applied. Students understand that the unknown angle is an actual,
measureable angle; they simply need to find the value that makes each equation true. They set up the equations based
on the angle facts they learned in Grade 4. The problems presented are not as simple as in Grade 4 because diagrams
incorporate angle facts in combination, rather than in isolation. Encourage students to verify their answers by
measuring relevant angles in each diagramโall diagrams are drawn to scale.
Classwork
Opening (5 minutes)
Discuss the ways in which angles are named and notated.
What do you notice about the three figures on the next page? What is the same about all three figures; what
is different?
There are three angles that appear to be the same measurement but are notated differently.
What is a likely implication of the three different kinds of notation?
They indicate the different ways of labeling or identifying the angle.
Students are familiar with addressing Figure 1 as ๐ and having a measurement of ๐ยฐ and addressing Figure 2 as angle ๐ด.
Elicit this from students and say that in a case like Figure 1, the angle is named by the measure of the arc, and in a case
like Figure 2, the angle is named by the single letter.
In a case like Figure 3, we use three letters when we name the angle. Why use three points to name an angle?
In a figure where several angles share the same vertex, naming a particular angle by the vertex point is
not sufficient information to distinguish that angle. Two additional points, one belonging to each side
of the intended angle, are necessary to identify it.
Encourage students to use both multiple forms of angle notation in the table to demonstrate each angle relationship.
NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
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aยฐ
bยฐ
B
A
C
D
bยฐaยฐ
E
D
F
G
C
C
A B D
aยฐ bยฐ
Figure 1 Figure 2 Figure 3
Naming by the arc. Naming by the vertex. Naming by three points.
โ ๐ โ ๐ด โ ๐ถ๐ด๐ท or โ ๐ท๐ด๐ถ
Recall the definitions of adjacent and vertical and the facts regarding angles on a line and angles at a point. If an
abbreviation exists, students should include the abbreviation of the angle fact under the name of each relationship. In
the Angle Fact column, students write the definitions and practice the different angle notations when describing the
relationship in the angle fact.
Note: The angles on a line fact applies to two or more angles.
Angle Facts and Definitions
Name of Angle
Relationship Angle Fact Diagram
Adjacent Angles
Two angles, โ ๐ฉ๐จ๐ช and โ ๐ช๐จ๐ซ with a
common side ๐จ๐ชโโโโ โ, are adjacent angles if ๐ช
belongs to the interior of โ ๐ฉ๐จ๐ซ.
Angles ๐ and ๐ are adjacent angles;
โ ๐ฉ๐จ๐ช and โ ๐ช๐จ๐ซ are adjacent angles.
Vertical Angles
(vert. โ s)
Two angles are vertical angles (or
vertically opposite angles) if their sides
form two pairs of opposite rays.
๐ = ๐
๐โ ๐ซ๐ช๐ญ = ๐โ ๐ฎ๐ช๐ฌ
Angles on a Line
(โ s on a line)
The sum of the measures of two angles
that share a ray and form a line is ๐๐๐ยฐ.
๐ + ๐ = ๐๐๐
๐โ ๐จ๐ฉ๐ช + ๐โ ๐ช๐ฉ๐ซ = ๐๐๐ยฐ
Angles at a Point
(โ s at a point)
The measure of all angles formed by three
or more rays with the same vertex is ๐๐๐ยฐ.
๐ + ๐ + ๐ = ๐๐๐
๐โ ๐ฉ๐จ๐ช + ๐โ ๐ช๐จ๐ซ + ๐โ ๐ซ๐จ๐ฉ = ๐๐๐ยฐ
bยฐA A
C
D
aยฐbยฐ
B
A
C
D
cยฐ
NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
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Opening Exercise (4 minutes)
Opening Exercise
Use the diagram to complete the chart.
Name the angles that are โฆ
Vertical โ ๐จ๐ฌ๐ช and โ ๐ฉ๐ฌ๐ซ, โ ๐ช๐ฌ๐ฉ and
โ ๐ซ๐ฌ๐จ
Adjacent
Answers include:
โ ๐จ๐ฌ๐ช and โ ๐ช๐ฌ๐ญ
โ ๐ช๐ฌ๐ญ and โ ๐ญ๐ฌ๐ฉ
Angles on a line
Answers include:
โ ๐ฉ๐ฌ๐ซ, โ ๐ซ๐ฌ๐ฎ, and โ ๐ฎ๐ฌ๐จ
โ ๐จ๐ฌ๐ช, โ ๐ช๐ฌ๐ญ, and โ ๐ญ๐ฌ๐ฉ
Angles at a
point
โ ๐จ๐ฌ๐ช, โ ๐ช๐ฌ๐ญ, โ ๐ญ๐ฌ๐ฉ, โ ๐ฉ๐ฌ๐ซ,
โ ๐ซ๐ฌ๐ฎ, โ ๐ฎ๐ฌ๐จ
Example 1 (4 minutes)
Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have students
verify their answers by measuring the unknown angle with a protractor.
Example 1
Estimate the measurement of ๐.
In a complete sentence, describe the angle relationship in the diagram.
โ ๐ฉ๐จ๐ช and โ ๐ช๐จ๐ซ are angles on a line and their measures
have a sum of ๐๐๐ยฐ.
Write an equation for the angle relationship shown in the
figure and solve for ๐. Then, find the measures of โ ๐ฉ๐จ๐ช and
confirm your answers by measuring the angle with a
protractor.
๐ + ๐๐๐ = ๐๐๐
๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
๐ = ๐๐
๐โ ๐ฉ๐จ๐ช = ๐๐ยฐ
NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
153
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Exercise 1 (4 minutes)
Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have students
verify their answers by measuring the unknown angle with a protractor.
Exercise 1
In a complete sentence, describe the angle relationship in the diagram.
โ ๐ฉ๐จ๐ช, โ ๐ช๐จ๐ซ, and โ ๐ซ๐จ๐ฌ are angles on a line and their measures have a sum of ๐๐๐ยฐ.
Find the measurements of โ ๐ฉ๐จ๐ช and
โ ๐ซ๐จ๐ฌ.
๐๐ + ๐๐ + ๐๐ = ๐๐๐
๐๐ + ๐๐ = ๐๐๐
๐๐ + ๐๐ โ ๐๐ = ๐๐๐ โ ๐๐
(๐
๐) (๐๐) = (
๐
๐) (๐๐)
๐ = ๐๐
๐โ ๐ฉ๐จ๐ช = ๐(๐๐ยฐ) = ๐๐ยฐ
๐โ ๐ซ๐จ๐ฌ = ๐(๐๐ยฐ) = ๐๐ยฐ
Example 2 (4 minutes)
Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have students
verify their answers by measuring the unknown angle with a protractor.
Example 2
In a complete sentence, describe the angle relationship in the diagram.
โ ๐จ๐ฌ๐ณ and โ ๐ณ๐ฌ๐ฉ are angles on a line and their
measures have a sum of ๐๐๐ยฐ. โ ๐จ๐ฌ๐ณ and โ ๐ฒ๐ฌ๐ฉ are
vertical angles and are of equal measurement.
Write an equation for the angle relationship shown
in the figure and solve for ๐ and ๐. Find the
measurements of โ ๐ณ๐ฌ๐ฉ and โ ๐ฒ๐ฌ๐ฉ.
๐ = ๐๐๐ยฐ; ๐โ ๐ฒ๐ฌ๐ฉ = ๐๐๐ยฐ (or vert. โ s are =)
๐ + ๐๐๐ = ๐๐๐
๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
๐ = ๐๐
๐โ ๐ณ๐ฌ๐ฉ = ๐๐ยฐ
NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
154
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Exercise 2 (4 minutes)
Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have students
verify their answers by measuring the unknown angle with a protractor.
Exercise 2
In a complete sentence, describe the angle relationships in the diagram.
โ ๐ฑ๐ฌ๐ต and โ ๐ต๐ฌ๐ด are adjacent angles and, when added together, are the
measure of โ ๐ฑ๐ฌ๐ด; โ ๐ฑ๐ฌ๐ด and โ ๐ฒ๐ฌ๐ณ are vertical angles and are of equal
measurement.
Write an equation for the angle relationship shown in the figure and solve
for ๐.
๐๐ + ๐๐ = ๐๐
๐๐ + ๐๐ โ ๐๐ = ๐๐ โ ๐๐
๐๐ = ๐๐
(๐
๐)๐๐ = ๐๐(
๐
๐)
๐ = ๐๐
Example 3 (4 minutes)
Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have students
verify their answers by measuring the unknown angle with a protractor.
Example 3
In a complete sentence, describe the angle relationships in the diagram.
โ ๐ฎ๐ฒ๐ฌ, โ ๐ฌ๐ฒ๐ญ, and โ ๐ฎ๐ฒ๐ญ are angles at a point and their measures have a
sum of ๐๐๐ยฐ.
Write an equation for the angle relationship shown in the figure and solve
for ๐. Find the measurement of โ ๐ฌ๐ฒ๐ญ and confirm your answers by
measuring the angle with a protractor.
๐ + ๐๐ + ๐๐๐ = ๐๐๐
๐ + ๐๐๐ = ๐๐๐
๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
๐ = ๐๐๐
๐โ ๐ฌ๐ฒ๐ญ = ๐๐๐ยฐ
E
L
J
K
M
3xยฐ
N
16ยฐ
85ยฐ
E
xยฐ
F
G K
135ยฐ
NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
155
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exercise 3 (4 minutes)
Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have students
verify their answers by measuring the unknown angle with a protractor.
Exercise 3
In a complete sentence, describe the angle relationships in the
diagram.
โ ๐ฌ๐จ๐ฏ, โ ๐ฎ๐จ๐ฏ, โ ๐ฎ๐จ๐ญ, and โ ๐ญ๐จ๐ฌ are angles at a point and
their measures sum to ๐๐๐ยฐ.
Find the measurement of โ ๐ฎ๐จ๐ฏ.
(๐ + ๐) + ๐๐ + ๐๐๐ + ๐๐๐ = ๐๐๐
๐ + ๐ + ๐๐ + ๐๐๐ + ๐๐๐ = ๐๐๐
๐ = ๐๐
๐โ ๐ฎ๐จ๐ฏ = (๐๐ + ๐)ยฐ = ๐๐ยฐ
Example 4 (5 minutes)
List pairs of angles whose measurements are in a ratio of 2 โถ 1.
Examples include: 90ยฐ and 45ยฐ, 60ยฐ and 30ยฐ, 150ยฐ and 75ยฐ.
What does it mean for the ratio of the measurements of two angles to be 2 โถ 1?
The measurement of one angle is two times the measure of the other
angle. If the smaller angle is defined as ๐ฅยฐ, then the larger angle is 2๐ฅยฐ.
If the larger angle is defined as ๐ฅยฐ, then the smaller angle is 1
2๐ฅยฐ.
Based on the following figure, which angle relationship(s) can be utilized to find the measure of an obtuse and
acute angle?
Any adjacent angle pair are on a line and have a sum of 180ยฐ.
Students describe the angle relationship in the question and set up and solve an equation that models it. Have students
verify their answers by measuring the unknown angle with a protractor.
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
๐ โถ ๐. In a complete sentence, describe the angle relationships in the
diagram.
The measurement of an obtuse angle is twice the measurement of an acute
angle in the diagram.
Scaffolding:
Students may find it helpful to
highlight the pairs of equal
vertical angles. MP.8
xยฐ2xยฐ
NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
156
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Label the diagram with expressions that describe this relationship. Write an equation that models the angle relationship
and solve for ๐. Find the measurements of the acute and obtuse angles.
๐๐ + ๐๐ = ๐๐๐
๐๐ = ๐๐๐
(๐
๐) (๐๐) = (
๐
๐) (๐๐๐)
๐ = ๐๐
Acute angle = ๐๐ยฐ
Obtuse angle = ๐๐ยฐ = ๐(๐๐ยฐ) = ๐๐๐ยฐ
Exercise 4 (4 minutes)
Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have students
verify their answers by measuring the unknown angle with a protractor.
Exercise 4
The ratio of ๐โ ๐ฎ๐ญ๐ฏ to ๐โ ๐ฌ๐ญ๐ฏ is ๐ โถ ๐. In a complete sentence, describe the angle relationships in the diagram.
The measurement of โ ๐ฎ๐ญ๐ฏ is ๐
๐ the measurement of โ ๐ฌ๐ญ๐ฏ; The measurements of โ ๐ฎ๐ญ๐ฏ and โ ๐ฌ๐ญ๐ฏ have a sum of ๐๐ยฐ.
Find the measures of โ ๐ฎ๐ญ๐ฏ and โ ๐ฌ๐ญ๐ฏ.
๐๐ + ๐๐ = ๐๐
๐๐ = ๐๐
(๐
๐) (๐๐) = (
๐
๐) (๐๐)
๐ = ๐๐
๐โ ๐ฎ๐ญ๐ฏ = ๐(๐๐ยฐ) = ๐๐ยฐ
๐โ ๐ฌ๐ญ๐ฏ = ๐(๐๐ยฐ) = ๐๐ยฐ
Relevant Vocabulary
ADJACENT ANGLES: Two angles โ ๐ฉ๐จ๐ช and โ ๐ช๐จ๐ซ with a common side ๐จ๐ชโโโโ โ are adjacent angles if ๐ช belongs to the interior of
โ ๐ฉ๐จ๐ซ.
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 ๐๐๐ยฐ.
ANGLES AT A POINT: The sum of the measures of adjacent angles at a point is ๐๐๐ยฐ.
Exit Ticket (3 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
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Name ___________________________________________________ Date____________________
Lesson 10: Angle Problems and Solving Equations
Exit Ticket
In a complete sentence, describe the relevant angle relationships in the
following diagram. That is, describe the angle relationships you could use to
determine the value of ๐ฅ.
Use the angle relationships described above to write an equation to solve for ๐ฅ. Then, determine the measurements of
โ ๐ฝ๐ด๐ป and โ ๐ป๐ด๐บ.
E
30ยฐ
3xยฐ5xยฐ
G
FJ
H
K
A
NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
158
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Exit Ticket Sample Solutions
In a complete sentence, describe the relevant angle relationships in the
following diagram. That is, describe the angle relationships you could use to
determine the value of ๐.
โ ๐ฒ๐จ๐ฌ and โ ๐ฌ๐จ๐ญ are adjacent angles whose measurements are equal to
โ ๐ฒ๐จ๐ญ; โ ๐ฒ๐จ๐ญ and โ ๐ฑ๐จ๐ฎ are vertical angles and are of equal measurement.
Use the angle relationships described above to write an equation to solve for
๐. Then, determine the measurements of โ ๐ฑ๐จ๐ฏ and โ ๐ฏ๐จ๐ฎ.
๐๐ + ๐๐ = ๐๐ + ๐๐
๐๐ = ๐๐๐
(๐
๐) (๐๐) = (
๐
๐) (๐๐๐)
๐ = ๐๐
๐โ ๐ฑ๐จ๐ฏ = ๐(๐๐ยฐ) = ๐๐ยฐ
๐โ ๐ฏ๐จ๐ฎ = ๐(๐๐ยฐ) = ๐๐ยฐ
Problem Set Sample Solutions
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 โ ๐ซ๐จ๐ฌ.
One possible response: โ ๐ช๐จ๐ซ, โ ๐ซ๐จ๐ฌ, and โ ๐ญ๐จ๐ฌ are
angles on a line and their measures sum to ๐๐๐ยฐ.
๐๐ + ๐ + ๐๐ = ๐๐๐
๐ + ๐๐๐ = ๐๐๐
๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
๐ = ๐๐
๐โ ๐ซ๐จ๐ฌ = ๐๐ยฐ
2. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurement
of โ ๐ธ๐ท๐น.
โ ๐ธ๐ท๐น, โ ๐น๐ท๐บ, and โ ๐บ๐ท๐ป are angles on a line and
their measures sum to ๐๐๐ยฐ.
๐ + ๐๐๐ + ๐ = ๐๐๐
๐๐ + ๐๐๐ = ๐๐๐
๐๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
๐๐ = ๐๐
(๐
๐)๐๐ = (
๐
๐)๐๐
๐ = ๐๐
๐โ ๐ธ๐ท๐น = ๐๐ยฐ
E
30ยฐ
3xยฐ5xยฐ
G
FJ
H
K
A
NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
159
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3. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurements
of โ ๐ช๐ธ๐ซ and โ ๐ฌ๐ธ๐ญ.
โ ๐ฉ๐ธ๐ช, โ ๐ช๐ธ๐ซ, โ ๐ซ๐ธ๐ฌ, โ ๐ฌ๐ธ๐ญ, and โ ๐ญ๐ธ๐ฎ are angles on a line and their measures sum to ๐๐๐ยฐ.
๐๐ + ๐๐ + ๐๐๐ + ๐๐ + ๐๐ = ๐๐๐
๐๐ + ๐๐๐ = ๐๐๐
๐๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
๐๐ = ๐๐
(๐
๐)๐๐ = (
๐
๐)๐๐
๐ = ๐๐
๐โ ๐ช๐ธ๐ซ = ๐(๐๐ยฐ) = ๐๐ยฐ
๐โ ๐ฌ๐ธ๐ญ = ๐(๐๐ยฐ) = ๐๐ยฐ
4. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of ๐.
All of the angles in the diagram are angles at a point and their
measures sum to ๐๐๐ยฐ.
๐(๐ + ๐๐) = ๐๐๐
๐๐ + ๐๐๐ = ๐๐๐
๐๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
๐๐ = ๐๐
(๐
๐)๐๐ = (
๐
๐)๐๐
๐ = ๐๐
5. In a complete sentence, describe the relevant angle relationships in
the following diagram. Find the measures of ๐ and ๐.
โ ๐ช๐ฒ๐ฌ, โ ๐ฌ๐ฒ๐ซ, and โ ๐ซ๐ฒ๐ฉ are angles on a line and their measures sum
to ๐๐๐ยฐ. Since โ ๐ญ๐ฒ๐จ and โ ๐จ๐ฒ๐ฌ form a straight angle and the
measurement of โ ๐ญ๐ฒ๐จ is ๐๐ยฐ, โ ๐จ๐ฒ๐ฌ is ๐๐ยฐ, making โ ๐ช๐ฒ๐ฌ and โ ๐จ๐ฒ๐ช
form a right angle and their measures have a sum of ๐๐ยฐ.
๐ + ๐๐ + ๐๐ = ๐๐๐
๐ + ๐๐๐ = ๐๐๐
๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
๐ = ๐๐
(๐๐) + ๐ = ๐๐
๐๐ โ ๐๐ + ๐ = ๐๐ โ ๐๐
๐ = ๐๐
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NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
160
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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6. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures of ๐
and ๐.
โ ๐ฌ๐จ๐ฎ and โ ๐ญ๐จ๐ฒ are vertical angles and are of equal measurement.
โ ๐ฌ๐จ๐ฎ and โ ๐ฎ๐จ๐ซ form a right angle and their measures have a sum
of ๐๐ยฐ.
๐๐ + ๐๐ = ๐๐
๐๐ + ๐๐ โ ๐๐ = ๐๐ โ ๐๐
๐๐ = ๐๐
(๐
๐)๐๐ = (
๐
๐)๐๐
๐ = ๐๐
๐๐ = ๐๐
(๐
๐)๐๐ = (
๐
๐)๐๐
๐ = ๐๐
7. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures
of โ ๐ช๐จ๐ซ and โ ๐ซ๐จ๐ฌ.
โ ๐ช๐จ๐ซ and โ ๐ซ๐จ๐ฌ form a right angle and their measures have a sum of
๐๐ยฐ.
(๐
๐๐ + ๐๐) + ๐๐ = ๐๐
๐
๐๐ + ๐๐ = ๐๐
๐
๐๐ + ๐๐ โ ๐๐ = ๐๐ โ ๐๐
๐
๐๐ = ๐๐
(๐
๐)๐
๐๐ = ๐๐(
๐
๐)
๐ = ๐๐
๐โ ๐ช๐จ๐ซ =๐
๐(๐๐ยฐ) + ๐๐ยฐ = ๐๐ยฐ
๐โ ๐ซ๐จ๐ฌ = ๐(๐๐ยฐ) = ๐๐ยฐ
8. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of
โ ๐ช๐ธ๐ฎ.
โ ๐ซ๐ธ๐ฌ and โ ๐ช๐ธ๐ญ are vertical angles and are of equal measurement.
โ ๐ช๐ธ๐ฎ and โ ๐ฎ๐ธ๐ญ are adjacent angles and their measures sum to the
measure of โ ๐ช๐ธ๐ญ.
๐๐ + ๐๐ = ๐๐๐
๐๐ + ๐๐ โ ๐๐ = ๐๐๐ โ ๐๐
๐๐ = ๐๐
(๐
๐)๐๐ = (
๐
๐)๐๐
๐ = ๐๐
๐โ ๐ช๐ธ๐ฎ = ๐(๐๐ยฐ) = ๐๐ยฐ
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NYS COMMON CORE MATHEMATICS CURRICULUM 7โข3 Lesson 10
Lesson 10: Angle Problems and Solving Equations
161
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
9. The ratio of the measures of a pair of adjacent angles on a line is ๐ โถ ๐.
a. Find the measures of the two angles.
โ ๐ = ๐๐, โ ๐ = ๐๐
๐๐ + ๐๐ = ๐๐๐
๐๐ = ๐๐๐
(๐
๐)๐๐ = (
๐
๐)๐๐๐
๐ = ๐๐
โ ๐ = ๐(๐๐ยฐ) = ๐๐ยฐ
โ ๐ = ๐(๐๐ยฐ) = ๐๐๐ยฐ
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 ๐ โถ ๐ โถ ๐.
a. Find the measures of the three angles.
โ ๐ = ๐๐, โ ๐ = ๐๐, โ ๐ = ๐๐
๐๐ + ๐๐ + ๐๐ = ๐๐๐
๐๐๐ = ๐๐๐
(๐
๐๐)๐๐๐ = (
๐
๐๐)๐๐๐
๐ = ๐๐
โ ๐ = ๐(๐๐ยฐ) = ๐๐ยฐ
โ ๐ = ๐(๐๐ยฐ) = ๐๐ยฐ
โ ๐ = ๐(๐๐ยฐ) = ๐๐ยฐ
b. Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle.
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