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Complementary and
Supplementary Angles
Section 2.3
Objective
• Find measures of complementary and
supplementary angles.
Key Vocabulary
•
•
•
•
Complementary angles
Supplementary angles
Adjacent angles
Theorem
Theorems
• 2.1 Congruent Complements Theorem
• 2.2 Congruent Supplements Theorem
Pairs of Angles
• Adjacent Angles – two angles that lie in the same
plane, have a common vertex and a common side, but
no common interior points
• Examples
• Adjacent angles: ∢1 and ∢2 are adjacent angles
• NOT adjacent angles: ∢3 and ∢ABC are not adjacent
angles
These angles are adjacent.
80º
45º
35º
55º
130º
85º
20º
50º
These angles are NOT adjacent.
100º
50º
35º
35º
55º
45º
Example 1:
Tell whether the numbered angles are adjacent or nonadjacent.
b.
a.
c.
SOLUTION
a. Because the angles do not share a common vertex or side,
1 and 2 are nonadjacent.
b. Because the angles share a common vertex and side,
and they do not have any common interior points, 3
and 4 are adjacent.
c. Although 5 and 6 share a common vertex,
they do not share a common side. Therefore,
5 and 6 are nonadjacent.
Angle Pair Relationships
• Two Types
• Complementary Angles
• Supplementary Angles
• Remember, angle measures are real
numbers, so the operations for real
numbers and algebra can apply to
angles.
Angle Pair Relationships
• Complementary Angles – two angles whose
measures have a sum of 90º
• Examples: ∢1 and ∢2 are complementary; ∢A is
complementary to ∢B
Complementary Angles
Two angles are complementary angles if the sum of
their measurements is 90˚. Each angle is the
complement of the other. Complementary angles can be
adjacent or nonadjacent.
4
1
3
2
complementary
adjacent
complementary
nonadjacent
Angle Pair Relationships
• Supplementary Angles – two angles whose
measures have a sum of 180º
• Examples: ∢3 and ∢4 are supplementary; ∢P
and ∢Q are supplementary
Supplementary Angles
Two angles are supplementary angles if the sum of their
measurements is 180˚. Each angle is the supplement of the
other. Supplementary angles can be adjacent or
nonadjacent.
7
5
8
6
supplementary
adjacent
supplementary
nonadjacent
Identifying Complementary and
Supplementary Angles
• Complementary angles make a
Corner of a piece of paper.
• Supplementary angles make up
the Sides of a piece of paper.
Example 2:
State whether the two angles are complementary,
supplementary, or neither.
SOLUTION
The angle showing 4:00
has a measure of 120˚
and the angle showing
10:00 has a measure of
60˚.
Because the sum of these two
measures is 180˚, the angles are
supplementary.
Example 3:
Determine whether the angles are complementary, supplementary, or
neither.
b.
a.
c.
SOLUTION
a. Because 22° + 158° = 180°, the angles are supplementary.
b. Because 15° + 85° = 100°, the angles are neither
complementary nor supplementary.
c. Because 55° + 35° = 90°, the angles are
complementary.
Your Turn:
Determine whether the angles are complementary,
supplementary, or neither.
1.
ANSWER
neither
ANSWER
complementary
ANSWER
supplementary
2.
3.
Example 4:
Find the angle measure.
Given that  A is a complement of C and m A = 47˚,
find mC.
SOLUTION
mC = 90˚ – m A
= 90˚ – 47˚
= 43˚
Example 5:
Find the angle measure.
Given that P is a supplement of R and mR = 36˚, find
mP.
SOLUTION
mP = 180˚ – mR
= 180 ˚ – 36˚
= 144˚
Your Turn:
1. B is a complement of D, and mD = 79°. Find mB.
ANSWER
11°
2. G is a supplement of H, and mG = 115°. Find mH.
ANSWER
65°
Example 6:
W and  Z are complementary. The measure of  Z is 5
times the measure of W. Find m W
SOLUTION
Because the angles are complementary,
m W + m  Z = 90˚.
But m  Z = 5( m W ),
so m W + 5( m W) = 90˚.
Because 6(m W) = 90˚,
you know that m W = 15˚.
Theorems
• We use undefined terms, definitions,
postulates, and algebraic properties of
equality to prove that other statements or
conjectures are true. Once a statement or
conjecture has been shown to be true, it is
called a theorem.
• Once proven true, a theorem can be used
like a definition or postulate to justify other
statements or conjectures.
• Thus, a theorem is a true statement that
follows from other true statements.
Complement Theorem
Theorem 2.1 (Complement Theorem)
If the noncommon sides of two adjacent
angles form a right angle, then the angles
are complementary angles.
m∠1 + m∠2 = 90
1
2
Supplement Theorem
Theorem 2.2 (Supplement Theorem)
If two angles form a linear pair, then they are
supplementary angles.
m∠1 + m∠2 = 180
1
2
Example 7:
7 and 8 are supplementary, and
8 and 9 are supplementary.
Name a pair of congruent angles.
Explain your reasoning.
SOLUTION
7 and 9 are both supplementary to 8. So, by the
Congruent supplements Theorem, 7  9.
Your Turn:
In the diagram, m10 + m11 = 90°, and
m11 + m12 = 90°.
Name a pair of congruent
angles. Explain your reasoning.
ANSWER
10  12; 10 and 12 are both
complementary to 11, so 10  12 by
the Congruent Complements Theorem.
Practice Time!
Directions:
Identify each pair of angles as
supplementary, complementary,
or neither.
#1
120º
60º
#1
120º
60º
Supplementary Angles
#2
30º
60º
#2
30º
60º
Complementary Angles
#3
40º
60º
#3
60º
40º
neither
#4
135º
45º
#4
135º
45º
Supplementary Angles
#5
25º
65º
#5
25º
65º
Complementary Angles
#6
90º
50º
#6
90º
50º
neither
Directions:
Determine the missing angle.
#1
?º
45º
#1
135º
45º
#2
?º
65º
#2
25º
65º
#3
?º
50º
#3
130º
50º
#4
?º
40º
#4
50º
40º
Joke Time
• Why did the geometry student get so
excited after they finished a jigsaw puzzle
in only 6 months?
• Because on the box it said from 2-4 years.
• Why did the geometry student climb the
chain-link fence?
• To see what was on the other side.
• How did the geometry student die drinking
milk?
• The cow fell on them.
Assignment
• Section 2-3, pg. 70-73: #1-37 odd, 41-53
odd