Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
7โข6
Lesson 3: Solving for Unknown Angles Using Equations Student Outcomes ๏ง
Students solve for unknown angles in word problems and in diagrams involving all learned angle facts.
Classwork Opening Exercise (5 minutes)
Scaffolding: Encourage students to redraw parts of the diagram to emphasize relationships. For example, line ๐ด๐ต and ray ๐๐ธ could be redrawn to see the relationship ๐ฆยฐ and the angle whose measure is 134ยฐ.
Opening Exercise Two lines meet at a point that is also a vertex of an angle; the measurement of โ ๐จ๐ถ๐ญ is ๐๐๐ยฐ. Set up and solve an equation to find the values of ๐ and ๐. Are your answers reasonable? How do you know? ๐๐ + ๐ + ๐๐ = ๐๐๐
โ s add
๐ + ๐๐๐ = ๐๐๐
14ยฐ
๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
O
C
๐ = ๐๐ ๐ + ๐๐๐ = ๐๐๐
A
xยฐ 134ยฐ
โ s on a line
๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
D
yยฐ
B
E
๐ = ๐๐ The answers are reasonable because the angle marked ๐ยฐ appears to be approximately half the measurement of a right angle, and the angle marked ๐ยฐ appears to be approximately double in measurement of โ ๐จ๐ถ๐ช.
F
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. Encourage students to note the appropriate angle fact abbreviation for any step that depends on an angle relationship.
Scaffolding: Remind students that a full rotation or turn through a circle is 360ยฐ.
Example 1 (4 minutes) Example 1 Set up and solve an equation to find the value of ๐. ๐ + ๐๐ + ๐๐๐ = ๐๐๐
โ s at a point
๐ + ๐๐๐ = ๐๐๐
xยฐ
๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
123ยฐ
๐ = ๐๐๐
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
A circular protractor may help to demonstrate this.
32 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
7โข6
Exercise 1 (4 minutes) Exercise 1 Five rays meet at a common endpoint. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of ๐. The sum of angles at a point is ๐๐๐ยฐ. ๐๐ + (๐๐ โ ๐๐) + ๐ + ๐๐๐ = ๐๐๐
โ s at a point
๐๐๐ + ๐ = ๐๐๐ ๐๐๐ โ ๐๐๐ + ๐ = ๐๐๐ โ ๐๐๐ ๐ = ๐๐
Example 2 (4 minutes) Example 2 Four rays meet at a common endpoint. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of ๐. Find the measurements of โ ๐ฉ๐จ๐ช and โ ๐ซ๐จ๐ฌ. The sum of the degree measurements of โ ๐ฉ๐จ๐ช, โ ๐ช๐จ๐ซ, โ ๐ซ๐จ๐ฌ and the arc that measures ๐๐๐ยฐ is ๐๐๐ยฐ. ๐ + ๐๐ + ๐๐ + ๐๐๐ = ๐๐๐
C B
xยฐ D
โ s at a point
๐๐ + ๐๐๐ = ๐๐๐ ๐๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐ A
๐๐ = ๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐ ๐ ๐ ๐ = ๐๐
5xยฐ
204ยฐ E
The measurement of โ ๐ฉ๐จ๐ช: ๐๐ยฐ The measurement of โ ๐ซ๐จ๐ฌ: ๐(๐๐)ยฐ = ๐๐ยฐ
Exercise 2 (4 minutes) Exercise 2 Four rays meet at a common endpoint. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of ๐. Find the measurement of โ ๐ช๐จ๐ซ. โ ๐ฉ๐จ๐ช, โ ๐ช๐จ๐ซ, โ ๐ซ๐จ๐ฌ, and โ ๐ฌ๐จ๐ฉ are angles at a point and sum to ๐๐๐ยฐ. ๐๐ + ๐๐ + ๐๐๐ + ๐๐ = ๐๐๐
โ s at a point
๐๐๐ + ๐๐๐ = ๐๐๐ ๐๐๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐ ๐๐๐ = ๐๐๐ ๐ ๐ ( ) ๐๐๐ = ( ) ๐๐๐ ๐๐ ๐๐ ๐ = ๐๐
12xยฐ
B
A 60ยฐ
C
E
3xยฐ D
The measurement of โ ๐ช๐จ๐ซ: ๐(๐๐)ยฐ = ๐๐ยฐ
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
33 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
7โข6
Example 3 (4 minutes) Example 3 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 ๐. Find the measurements of โ ๐ฉ๐จ๐ช and โ ๐ฉ๐จ๐ฏ. โ ๐ซ๐จ๐ฌ is formed by adjacent angles โ ๐ฌ๐จ๐ญ and โ ๐ญ๐จ๐ซ; the measurement of โ ๐ซ๐จ๐ฌ is equal to the sum of the measurements of the adjacent angles. This is also true for the measurement of โ ๐ช๐จ๐ฏ, formed by adjacent angles โ ๐ช๐จ๐ฉ and โ ๐ฉ๐จ๐ฏ. โ ๐ช๐จ๐ฏ is vertically opposite from and equal in measurement to โ ๐ซ๐จ๐ฌ. ๐๐ + ๐๐ = ๐๐๐
โ ๐ซ๐จ๐ฌ, โ s add
๐๐ + ๐๐ = ๐๐
โ ๐ช๐จ๐ฏ, โ s add
F 30ยฐ
E
A
Vert. โ s
๐๐ = ๐๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐๐ ๐ ๐ ๐ = ๐๐
D
3xยฐ
H
5xยฐ C
B
The measurement of โ ๐ฉ๐จ๐ช: ๐(๐๐)ยฐ = ๐๐ยฐ The measurement of โ ๐ฉ๐จ๐ฏ: ๐(๐๐)ยฐ = ๐๐ยฐ
Exercise 3 (4 minutes) Exercise 3 Lines ๐จ๐ฉ and ๐ฌ๐ญ meet at a point which 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 ๐. Find the measurements of โ ๐ซ๐ฏ๐ญ and โ ๐จ๐ฏ๐ซ. The measurement of โ ๐จ๐ฏ๐ญ, formed by adjacent angles โ ๐จ๐ฏ๐ซ and โ ๐ซ๐ฏ๐ญ, is equal to the sum of the measurements of the adjacent angles. This is also true for the measurement of โ ๐ฌ๐ฏ๐ฉ, which is formed by adjacent angles โ ๐ฌ๐ฏ๐ช and โ ๐ช๐ฏ๐ฉ. โ ๐จ๐ฏ๐ญ is vertically opposite from and equal in measurement to โ ๐ฌ๐ฏ๐ฉ. ๐๐ + ๐ = ๐๐
โ ๐จ๐ฏ๐ญ, โ s add
๐๐ + ๐๐ = ๐๐๐
โ ๐ฌ๐ฏ๐ฉ, โ s add
๐๐ = ๐๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐๐ ๐ ๐ ๐ = ๐๐
Vert. โ s
The measurement of โ ๐ซ๐ฏ๐ญ: ๐๐ยฐ The measurement of โ ๐จ๐ฏ๐ซ: ๐(๐๐)ยฐ = ๐๐๐ยฐ
The following examples are designed to highlight MP.7 by helping students to see the connection between an angle diagram and the equation used to model it. Solving equations with variables on both sides is a topic in Grade 8. Teachers may choose to show the solution method if they so choose.
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
34 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
7โข6
Example 4 (6 minutes) Example 4 Two lines meet at a point. Set up and solve an equation to find the value of ๐. Find the measurement of one of the vertical angles.
3xยฐ
(x+30)ยฐ
Students use information in the figure and a protractor to solve for ๐ฅ.
MP.7
i)
Students measure a 30ยฐ angle as shown; the remaining portion of the angle must be ๐ฅยฐ (โ s add).
ii)
Students can use their protractor to find the measurement of ๐ฅยฐ and use this measurement to partition the other angle in the vertical pair.
As a check, students should substitute the measured ๐ฅ value into each expression and evaluate; each angle of the vertical pair should equal the other. Students can also use their protractor to measure each angle of the vertical angle pair. With a modified figure, students can write an algebraic equation that they have the skills to solve. ๐๐ = ๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐ ๐ ๐ ๐ = ๐๐
xยฐ xยฐ xยฐ
30ยฐ xยฐ
Vert. โ s
Measurement of each angle in the vertical pair: ๐(๐๐)ยฐ = ๐๐ยฐ
Extension: The algebra steps above are particularly helpful as a stepping-stone in demonstrating how to solve the equation that takes care of the problem in one step as follows: ๐๐ = ๐ + ๐๐
Vert. โ s
๐๐ โ ๐ = ๐ โ ๐ + ๐๐ ๐๐ = ๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐ ๐ ๐ ๐ = ๐๐ Measurement of each angle in the vertical pair: ๐(๐๐)ยฐ = ๐๐ยฐ
Students understand the first line of this solution because of their knowledge of vertical angles. In fact, the only line they are not familiar with is the second line of the solution, which is a skill that they learn in Grade 8. Showing students this solution is simply a preview.
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
35 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
7โข6
Exercise 4 (4 minutes) Exercise 4 Set up and solve an equation to find the value of ๐. Find the measurement of one of the vertical angles.
Students use information in the figure and a protractor to solve for ๐ฅ. i)
Students measure a 54ยฐ angle as shown; the remaining portion of the angle must be ๐ฅ (โ s add).
ii)
Students can use their protractors to find the measurement of ๐ฅ and use this measurement to partition the other angle in the vertical pair.
Students should perform a check as in Example 4 before solving an equation that matches the modified figure. ๐๐ = ๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐ ๐ ๐ ๐ = ๐๐
Vert. โ s
Measurement of each vertical angle: ๐(๐๐)ยฐ = ๐๐ยฐ
Extension: ๐ + ๐๐ = ๐๐
Vert. โ s
xยฐ 54ยฐ
xยฐ xยฐ
๐ โ ๐ + ๐๐ = ๐๐ โ ๐ ๐๐ = ๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐ ๐ ๐ ๐ = ๐๐
xยฐ xยฐ
Closing (1 minute) ๏ง
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.
๏ง
Check your answer by substituting and/or measuring to be sure it is correct.
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
36 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 3
7โข6
Lesson Summary Steps to Solving for Unknown Angles ๏ง
Identify the angle relationship(s).
๏ง
Set up an equation that will yield the unknown value.
๏ง
Solve the equation for the unknown value.
๏ง
Substitute the answer to determine the angle(s).
๏ง
Check and verify your answer by measuring the angle with a protractor.
Exit Ticket (5 minutes)
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
37 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
7โข6
Date
Lesson 3: Solving for Unknown Angles Using Equations Exit Ticket 1.
Two rays have a common endpoint on a line. Set up and solve an equation to find the value of ๐ง. Find the measurements of โ ๐ด๐๐ถ and โ ๐ท๐๐ต. C A 5zยฐ Y D
zยฐ B
2.
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 ๐ฅ. Find the measurements of โ ๐ถ๐ด๐ป and โ ๐ธ๐ด๐บ. G
E
4xยฐ
H xยฐ
B
A 160ยฐ
C
F
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
38 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
7โข6
Exit Ticket Sample Solutions 1.
Two rays have a common endpoint on a line. Set up and solve an equation to find the value of ๐. Find the measurements of โ ๐จ๐๐ช and โ ๐ซ๐๐ฉ. C ๐๐ + ๐๐ + ๐ = ๐๐๐
โ s on a line A
๐๐ + ๐๐ = ๐๐๐ ๐๐ + ๐๐ โ ๐๐ = ๐๐๐ โ ๐๐
5zยฐ
๐๐ = ๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐ ๐ ๐ ๐ = ๐๐
Y D
zยฐ
The measurement of โ ๐จ๐๐ช: ๐(๐๐)ยฐ = ๐๐ยฐ
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?
B
The measurement of โ ๐ซ๐๐ฉ: ๐๐ยฐ Scaffolded solutions:
2.
a.
Use the equation above.
b.
The angle marked ๐ยฐ, the right angle, and the angle with measurement ๐๐ยฐ are angles on a line. Their measurements sum to ๐๐๐ยฐ.
c.
The answers seem reasonable because once ๐๐ is substituted in for ๐, the measurement of โ ๐จ๐๐ช is ๐๐ยฐ, which is slightly smaller than a right angle, and the measurement of โ ๐ซ๐๐ฉ is ๐๐ยฐ, which is an acute angle.
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 ๐. Find the measurements of โ ๐ช๐จ๐ฏ and โ ๐ฌ๐จ๐ฎ. ๐๐ + ๐๐ + ๐ = ๐๐๐
vert. โ s
G
๐๐ + ๐๐ = ๐๐๐ ๐๐ + ๐๐ โ ๐๐ = ๐๐๐ โ ๐๐ ๐๐ = ๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐ ๐ ๐ ๐ = ๐๐
E
4xยฐ
H xยฐ
B
The measurement of โ ๐ช๐จ๐ฏ: ๐๐ยฐ
C
A 160ยฐ
The measurement of โ ๐ฌ๐จ๐ฎ: ๐(๐๐)ยฐ = ๐๐ยฐ
F
Problem Set Sample Solutions Set up and solve an equation for the unknown angle based on the relevant angle relationships in the diagram. Add labels to diagrams as needed to facilitate their solutions. List the appropriate angle fact abbreviation for any step that depends on an angle relationship. 1.
Two lines meet at a point. Set up and solve an equation to find the value of ๐. ๐ + ๐๐ = ๐๐
Vert. โ s
๐ + ๐๐ โ ๐๐ = ๐๐ โ ๐๐
72ยฐ
(x+15)ยฐ
๐ = ๐๐
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
39 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
7โข6
Three lines meet at a point. Set up and solve an equation to find the value of ๐. Is your answer reasonable? Explain how you know. Let ๐ = ๐.
Vert. โ s โ s on a line
๐๐ + ๐ + ๐๐ = ๐๐๐ ๐ + ๐๐๐ = ๐๐๐ ๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐ ๐ = ๐๐ Since ๐ = ๐, ๐ = ๐๐.
The answer seems reasonable since it is similar in magnitude to the ๐๐ยฐ angle.
3.
Two lines meet at a point that is also the endpoint of two rays. Set up and solve an equation to find the values of ๐ and ๐. ๐ + ๐๐ + ๐๐ = ๐๐๐
Scaffolding: Students struggling to organize their solution may benefit from prompts such as the following:
โ s on a line
๐ + ๐๐๐ = ๐๐๐ ๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐
๏ง Write an equation to model this situation. Explain how your equation describes the situation. Solve and interpret the solution. Is it reasonable?
32ยฐ
๐ = ๐๐ aยฐ
๐ + ๐ + ๐๐ = ๐๐๐
โ s on a line bยฐ
๐๐ + ๐ + ๐๐ = ๐๐๐ ๐ + ๐๐๐ = ๐๐๐ ๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐ ๐ = ๐๐
Scaffolded solutions:
4.
a.
Use the equation above.
b.
The angle marked ๐ยฐ, the angle with measurement ๐๐ยฐ, and the right angle are angles on a line. Their measurements sum to ๐๐๐ยฐ.
c.
The answers seem reasonable because once the values of ๐ and ๐ are substituted, it appears that the two angles (๐ยฐ and ๐ยฐ) form a right angle. We know those two angles should form a right angle because the angle adjacent to it is a right angle.
Three lines meet at a point that is also the endpoint of a ray. Set up and solve an equation to find the values of ๐ and ๐. ๐ + ๐๐ + ๐๐ = ๐๐๐
โ s on a line
๐ + ๐๐๐ = ๐๐๐ ๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐ ๐ = ๐๐ ๐ + ๐ + ๐๐ = ๐๐๐
โ s on a line
39ยฐ
๐ + ๐๐ + ๐๐ = ๐๐๐
yยฐ
๐ + ๐๐๐ = ๐๐๐
xยฐ
๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐ ๐ = ๐๐
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
40 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
7โข6
Two lines meet at a point. Find the measurement of one of the vertical angles. Is your answer reasonable? Explain how you know. ๐๐ = ๐๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐๐ ๐ ๐ ๐ = ๐๐
(x+104)ยฐ
vert. โ s
3xยฐ
Measurement of each vertical angle: ๐(๐๐)ยฐ = ๐๐๐ยฐ The answer seems reasonable because a rounded value of ๐๐ would make the numeric value of each expression ๐๐๐ and ๐๐๐, which are reasonably close for a check.
xยฐ
A solution can include a modified diagram, as shown, and the supporting algebra work.
xยฐ
104ยฐ
xยฐ
xยฐ
Solutions may also include the full equation and solution: ๐๐ = ๐ + ๐๐๐
Vert. โ s
๐๐ โ ๐ = ๐ โ ๐ + ๐๐๐ ๐๐ = ๐๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐๐ ๐ ๐ ๐ = ๐๐
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 ๐. Let ๐ยฐ and ๐ยฐ be the measurements of the indicated angles. ๐ + ๐๐ = ๐๐
15ยฐ
Vert. โ s
๐ + ๐๐ โ ๐๐ = ๐๐ โ ๐๐ ๐ = ๐๐ ๐ + ๐ = ๐๐
Complementary โ s
๐๐ + ๐ = ๐๐ ๐๐ โ ๐๐ + ๐ = ๐๐ โ ๐๐
๐ห ห
๐ห ห yยฐ
๐ = ๐๐ ๐ + ๐ = ๐๐๐
โ s on a line
๐๐ + ๐ = ๐๐๐ ๐๐ โ ๐๐ + ๐ = ๐๐๐ โ ๐๐ ๐ = ๐๐๐
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
41 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
7.
7โข6
Three adjacent angles are at a point. The second angle is ๐๐ยฐ more than the first, and the third angle is ๐๐ยฐ more than the second angle. a.
Find the measurements of all three angles. ๐ + (๐ + ๐๐) + (๐ + ๐๐ + ๐๐) = ๐๐๐
โ s at a point
๐๐ + ๐๐ = ๐๐๐ ๐๐ + ๐๐ โ ๐๐ = ๐๐๐ โ ๐๐ ๐๐ = ๐๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐๐ ๐ ๐ ๐ = ๐๐๐ Angle 1: ๐๐๐ยฐ Angle 2: ๐๐๐ยฐ + ๐๐ยฐ = ๐๐๐ยฐ Angle 3: ๐๐๐ยฐ + ๐๐ยฐ + ๐๐ยฐ = ๐๐๐ยฐ
b.
MP.2 & MP.7
Compare the expressions you used for the three angles and their combined expression. Explain how they are equal and how they reveal different information about this situation. By the commutative and associative laws, ๐ + (๐ + ๐๐) + (๐ + ๐๐ + ๐๐) is equal to (๐ + ๐ + ๐) + (๐๐ + ๐๐ + ๐๐), which is equal to ๐๐ + ๐๐. The first expression, ๐ + (๐ + ๐๐) + (๐ + ๐๐ + ๐๐), shows the sum of three unknown numbers, where the second is ๐๐ more than the first, and the third is ๐๐ more than the second. The expression ๐๐ + ๐๐ shows the sum of three times an unknown number with ๐๐.
8.
Four adjacent angles are on a line. The measurements of the four angles are four consecutive even numbers. Determine the measurements of all four angles. ๐ + (๐ + ๐) + (๐ + ๐) + (๐ + ๐) = ๐๐๐
โ s on a line
๐๐ + ๐๐ = ๐๐๐ ๐๐ + ๐๐ โ ๐๐ = ๐๐๐ โ ๐๐ ๐๐ = ๐๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐๐ ๐ ๐ ๐ = ๐๐
Scaffolding: Teachers may need to review the term consecutive for students to successfully complete Problem Set 8.
The four angle measures are ๐๐ยฐ, ๐๐ยฐ, ๐๐ยฐ, and ๐๐ยฐ.
9.
Three adjacent angles are at a point. The ratio of the measurement of the second angle to the measurement of the first angle is ๐: ๐. The ratio of the measurement of the third angle to the measurement of the second angle is ๐: ๐. Determine the measurements of all three angles. Let the smallest measure of the three angles be ๐๐ยฐ. Then, the measure of the second angle is ๐๐ยฐ, and the measure of the third angle is ๐๐ยฐ. ๐๐ + ๐๐ + ๐๐ = ๐๐๐
โ s at a point
๐๐๐ = ๐๐๐ ๐
๐
(๐๐) ๐๐๐ = (๐๐) ๐๐๐ ๐ = ๐๐ Angle 1: ๐(๐๐)ยฐ = ๐๐ยฐ Angle 2: ๐(๐๐)ยฐ = ๐๐๐ยฐ Angle 3: ๐(๐๐)ยฐ = ๐๐๐ยฐ
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
42 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 3
7โข6
10. Four lines meet at a point. Solve for ๐ and ๐ in the following diagram. ๐๐ + ๐๐ + ๐๐ = ๐๐๐
โ s on a line
๐๐ + ๐๐๐ = ๐๐๐ ๐๐ + ๐๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐ ๐๐ = ๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐ ๐ ๐ ๐ = ๐๐ ๐๐ = ๐๐
Vert. โ s
๐(๐๐) = ๐๐ ๐๐ = ๐๐ ๐ ๐ ( ) ๐๐ = ( ) ๐๐ ๐ ๐ ๐ = ๐๐
Lesson 3:
Solving for Unknown Angles Using Equations
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015
43 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.