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C HAPTER
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Complementary Angles
Here you’ll learn what complementary angles are and how to solve complementary angle problems.
What if you were given two angles of unknown size and were told they are complementary? How would you
determine their angle measures? After completing this Concept, you’ll be able to use the definition of complementary
angles to solve problems like this one.
Watch This
Watch this video beginning at around the 3:20 mark.
MEDIA
Click image to the left for more content.
http://www.youtube.com/watch?v=7iBc5bJdanI
Then watch the first part of this video.
MEDIA
Click image to the left for more content.
http://www.youtube.com/watch?v=rjOjwcV79HM
Guidance
Two angles are complementary if they add up to 90◦ . Complementary angles do not have to be congruent or next
to each other.
Example A
The two angles below are complementary. m� GHI = x. What is x?
Chapter 1. Complementary Angles
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Because the two angles are complementary, they add up to 90◦ . Make an equation.
x + 34◦ = 90◦
x = 56◦
Example B
The two angles below are complementary. Find the measure of each angle.
The two angles add up to 90◦ . Make an equation.
(8r + 9) + (7r + 6) = 90
(15r + 15) = 90
15r = 75
r=5
However, you need to find each angle. Plug r back into each expression.
m� GHI = 8(5◦ ) + 9◦ = 49◦
m� JKL = 7(5◦ ) + 6◦ = 41◦
Example C
Find the measure of an angle that is a complementary to � MRS if m� MRS is 70◦ .
Because complementary angles have to add up to 90◦ , the other angle must be 90◦ − 70◦ = 20◦ .
Vocabulary
Two angles are complementary if they add up to 90◦ .
Guided Practice
Find the measure of an angle that is complementary to � ABC if m� ABC is:
1. 45◦
2. 82◦
3. 19◦
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4. 12◦
Answers:
1. Because complementary angles have to add up to 90◦ , the other angle must be 90◦ − 45◦ = 45◦ .
2. Because complementary angles have to add up to 90◦ , the other angle must be 90◦ − 82◦ = 8◦ .
3. Because complementary angles have to add up to 90◦ , the other angle must be 90◦ − 19◦ = 71◦ .
4. Because complementary angles have to add up to 90◦ , the other angle must be 90◦ − 12◦ = 78◦ .
Interactive Practice
Practice
Find the measure of an angle that is complementary to � ABC if m� ABC is:
1.
2.
3.
4.
5.
6.
7.
4◦
89◦
54◦
32◦
27◦
(x + y)◦
z◦
←
→
Use the diagram below for exercises 8-9. Note that NK ⊥ IL .
8. Name two complementary angles.
9. If m� INJ = 63◦ , find m� KNJ.
For 10-11, determine if the statement is true or false.
10. Complementary angles add up to 180◦ .
11. Complementary angles are always 45◦ .
Chapter 1. Complementary Angles
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C HAPTER
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Supplementary Angles
Here you’ll learn what supplementary angles are and how to solve supplementary angle problems.
What if you were given two angles of unknown size and were told they are supplementary? How would you
determine their angle measures? After completing this Concept, you’ll be able to use the definition of supplementary
angles to solve problems like this one.
Watch This
Watch this video beginning at around the 3:20 mark.
MEDIA
Click image to the left for more content.
http://www.youtube.com/watch?v=7iBc5bJdanI
Then watch the second part of this video.
MEDIA
Click image to the left for more content.
http://www.youtube.com/watch?v=rjOjwcV79HM
Guidance
Two angles are supplementary if they add up to 180◦ . Supplementary angles do not have to be congruent or next
to each other.
Example A
The two angles below are supplementary. If m� MNO = 78◦ what is m� PQR?
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Set up an equation. However, instead of equaling 90◦ , now the sum is 180◦ .
78◦ + m� PQR = 180◦
m� PQR = 102◦
Example B
What are the measures of two congruent, supplementary angles?
Supplementary angles add up to 180◦ . Congruent angles have the same measure. So, 180◦ ÷ 2 = 90◦ , which means
two congruent, supplementary angles are right angles, or 90◦ .
Example C
Find the measure of an angle that is a supplementary to � MRS if m� MRS is 70◦ .
Because supplementary angles have to add up to 180◦ , the other angle must be 180◦ − 70◦ = 110◦ .
Vocabulary
Two angles are supplementary if they add up to 180◦ .
Guided Practice
Find the measure of an angle that is supplementary to � ABC if m� ABC is:
1.
2.
3.
4.
45◦
118◦
32◦
2◦
Answers:
1. Because supplementary angles have to add up to 180◦ , the other angle must be 180◦ − 45◦ = 135◦ .
2. Because supplementary angles have to add up to 180◦ , the other angle must be 180◦ − 118◦ = 62◦ .
3. Because supplementary angles have to add up to 180◦ , the other angle must be 180◦ − 32◦ = 148◦ .
4. Because supplementary angles have to add up to 180◦ , the other angle must be 180◦ − 2◦ = 178◦ .
Interactive Practice
Practice
Find the measure of an angle that is supplementary to � ABC if m� ABC is:
1.
2.
3.
4.
114◦
11◦
91◦
84◦
Chapter 2. Supplementary Angles
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5. 57◦
6. x◦
7. (x + y)◦
←
→
Use the diagram below for exercises 8-9. Note that NK ⊥ IL .
8. Name two supplementary angles.
9. If m� INJ = 63◦ , find m� JNL.
For exercise 10, determine if the statement is true or false.
10. Supplementary angles add up to 180◦
For 11-12, find the value of x.
11.
12.
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C HAPTER
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Linear Pairs
Here you’ll learn what linear pairs are and how to solve linear pair problems.
What if you were given two angles of unknown size and were told they form a linear pair? How would you determine
their angle measures? After completing this Concept, you’ll be able to use the definition of linear pair to solve
problems like this one.
Guidance
Two angles are adjacent if they have the same vertex, share a side, and do not overlap � PSQ and � QSR are adjacent.
A linear pair is two angles that are adjacent and whose non-common sides form a straight line. If two angles are a
linear pair, then they are supplementary (add up to 180◦ ). � PSQ and � QSR are a linear pair.
Example A
What is the measure of each angle?
These two angles are a linear pair, so they add up to 180◦ .
(7q − 46)◦ + (3q + 6)◦ = 180◦
10q − 40◦ = 180◦
10q = 220
q = 22
Chapter 3. Linear Pairs
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Plug in q to get the measure of each angle. m� ABD = 7(22◦ ) − 46◦ = 108◦ m� DBC = 180◦ − 108◦ = 72◦
Example B
Are � CDA and � DAB a linear pair? Are they supplementary?
The two angles are not a linear pair because they do not have the same vertex. They are supplementary because they
add up to 180◦ : 120◦ + 60◦ = 180◦ .
Example C
Find the measure of an angle that forms a linear pair with � MRS if m� MRS is 150◦ .
Because linear pairs have to add up to 180◦ , the other angle must be 180◦ − 150◦ = 30◦ .
Vocabulary
Two angles are adjacent if they have the same vertex, share a side, and do not overlap. A linear pair is two angles
that are adjacent and whose non-common sides form a straight line. If two angles are a linear pair, then they are
supplementary (add up to 180◦ ).
Guided Practice
←
→
Use the diagram below. Note that NK ⊥ IL .
1. Name one linear pair of angles.
2. What is m� INL?
3. What is m� LNK?
4. If m� INJ = 63◦ , find m� MNI.
Answers:
1. � MNL and � LNJ
2. 180◦
3. 90◦
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4. 180◦ − 63◦ = 117◦
Interactive Practice
Practice
For 1-5, determine if the statement is true or false.
1.
2.
3.
4.
5.
Linear pairs are congruent.
Adjacent angles share a vertex.
Adjacent angles overlap.
Linear pairs are supplementary.
Supplementary angles form linear pairs.
For exercise 6, find the value of x.
6.
Find the measure of an angle that forms a linear pair with � MRS if m� MRS is:
7.
8.
9.
10.
11.
12.
61◦
23◦
114◦
7◦
179◦
z◦
Chapter 3. Linear Pairs
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C HAPTER
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Vertical Angles
Here you’ll learn what vertical angles are and how to solve vertical angle problems.
What if you were given two angles of unknown size and were told they are vertical angles? How would you
determine their angle measures? After completing this Concept, you’ll be able to use the definition of vertical angles
to solve problems like this one.
Watch This
Watch the first part of this video.
MEDIA
Click image to the left for more content.
http://www.youtube.com/watch?v=z_O2Knid2XA
Then watch the third part of this video.
MEDIA
Click image to the left for more content.
http://www.youtube.com/watch?v=rjOjwcV79HM
Guidance
Vertical angles are two non-adjacent angles formed by intersecting lines. � 1 and � 3 are vertical angles and � 2 and
� 4 are vertical angles.
The Vertical Angles Theorem states that if two angles are vertical angles, then they are congruent.
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Example A
Find m� 1.
1 is vertical angles with 18◦ , so m� 1 = 18◦ .
�
Example B
If � ABC and � DEF are vertical angles and m� ABC = (4x + 10)◦ and m� DEF = (5x + 2)◦ , what is the measure of
each angle?
Vertical angles are congruent, so set the angles equal to each other and solve for x. Then go back to find the measure
of each angle.
4x + 10 = 5x + 2
x=8
So, m� ABC = m� DEF = (4(8) + 10)◦ = 42◦
Example C
True or false: vertical angles are always less than 90◦ .
This is false, you can have vertical angles that are more than 90◦ . Vertical angles are less than 180◦ .
Vocabulary
Vertical angles are two non-adjacent angles formed by intersecting lines.
Guided Practice
Find the value of x or y.
1.
2.
Chapter 4. Vertical Angles
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3.
Answers:
1. Vertical angles are congruent, so set the angles equal to each other and solve for x.
x + 16 = 4x − 5
3x = 21
x = 7◦
2. Vertical angles are congruent, so set the angles equal to each other and solve for y.
9y + 7 = 2y + 98
7y = 91
y = 13◦
3. Vertical angles are congruent, so set the angles equal to each other and solve for y.
11y − 36 = 63
11y = 99
y = 9◦
Interactive Practice
Practice
←
→
Use the diagram below for exercises 1-2. Note that NK ⊥ IL .
1. Name one pair of vertical angles.
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2. If m� INJ = 63◦ , find m� MNL.
For exercise 3, determine if the statement is true or false.
3. Vertical angles have the same vertex.
4. If � ABC and � DEF are vertical angles and m� ABC = (9x +1)◦ and m� DEF = (5x +29)◦ , what is the measure
of each angle?
5. If � ABC and � DEF are vertical angles and m� ABC = (8x +2)◦ and m� DEF = (2x +32)◦ , what is the measure
of each angle?
6. If � ABC and � DEF are vertical angles and m� ABC = (x + 22)◦ and m� DEF = (5x + 2)◦ , what is the measure
of each angle?
7. If � ABC and � DEF are vertical angles and m� ABC = (3x + 12)◦ and m� DEF = (7x)◦ , what is the measure
of each angle?
8. If � ABC and � DEF are vertical angles and m� ABC = (5x + 2)◦ and m� DEF = (x + 26)◦ , what is the measure
of each angle?
9. If � ABC and � DEF are vertical angles and m� ABC = (3x + 1)◦ and m� DEF = (2x + 2)◦ , what is the measure
of each angle?
10. If � ABC and � DEF are vertical angles and m� ABC = (6x − 3)◦ and m� DEF = (5x + 1)◦ , what is the measure
of each angle?
Chapter 4. Vertical Angles