PROVING STATEMENTS ABOUT
ANGLES
2.6
What you should learn
GOAL
1
Justify statements about congruent angles.
GOAL
2
Prove properties about special pairs of angles
Why you should learn it
Properties of special pairs of angles help you
determine angles in real-life applications, such as
design work.
PROVING STATEMENTS ABOUT
ANGLES
2.6
GOAL
1
PROPERTIES OF CONGRUENT ANGLES
VOCABULARY
PROPERTIES OF
ANGLE CONGRUENCE
Reflexive
Symmetric
Transitive
EXAMPLE 1
Extra Example 1
B
Given: 1  2, 3  4, 2  3
Prove: 1  4
1
2
4
A
Statements
3
C
Reasons
1. 1  2, 2  3
1. Given
2. 1  3
2. Transitive Prop. of 
3. 3  4
3. Given
4. 1  4
4. Transitive Prop. of 
EXAMPLE 2
Extra Example 2
Given: m1  63, 1  3, 3  4
1
3
Statements
Prove: m4  63
2
4
Reasons
1. 1  3, 3  4
1. Given
2. 1  4
2. Transitive Prop. of 
3. m1  m4
3. Def. of s
4. m1  63
4. Given
5. m4  63
5. Subs. Prop. of =
RIGHT ANGLE
CONGRUENCE THEOREM
All right angles are congruent.
EXAMPLE 3
Extra Example 3
Given: DAB, ABC are right angles; ABC  BCD
Prove: DAB  BCD
D
C
A
Statements
B
Reasons
1. DAB, ABC are right angles 1. Given
2. DAB  ABC
2. Right s are 
3. ABC  BCD
3. Given
4. DAB  BCD
4. Transitive Prop. of 
Checkpoint
Given: AFC, BFD are right angles, BFD  CFE
Prove: AFC  CFE
C
B
A
Statements
F
D
E
Reasons
1. AFC, BFD are right angles 1. Given
2. AFC  BFD
2. Right s are 
3. BFD  CFE
3. Given
4. AFC  CFE
4. Transitive Prop. of 
PROVING STATEMENTS ABOUT
ANGLES
2.6
GOAL
2
USING CONGRUENCE OF ANGLES
CONGRUENT SUPPLEMENTS THEOREM
Two angles supplementary to the same angle (or
congruent angles) are congruent
CONGRUENT COMPLEMENTS THEOREM
Two angles complementary to the same angle
(or congruent angles) are congruent
In proofs, these may be abbreviated
as  Supp. Thm. and  Comp. Thm.
EXAMPLE 4
Extra Example 4
Given: m1  24, m3  24,
Prove: 2  4
1 and 2 are complementary,
3 and 4 are complementary
1 2
4
3
Statements
Reasons
1. m1  24, m3  24,
1. Given
1 and 2 are complementary,
3 and 4 are complementary
2. m1  m3
3. 1  3
4. 2  4
2. Transitive Prop. of =
3. Def. of  s
4.  Complements Thm.
Checkpoint
1. In a diagram, 1 and 2 are supplementary and 2 and
3 are supplementary. Explain how to show that 1  3.
Using the definition of supplementary angles,
m1 m2  180 and m2  m3  180. So
m1 m2  m2  m3 by the transitive property of
equality. So m1  m3 by the subtraction property of
equality. Therefore, 1  3 by the definition of congruent
angles.
LINEAR PAIR POSTULATE
If two angles form a linear pair,
then they are supplementary.
EXAMPLE 5
Extra Example 5
In the diagram m1  60 and BFD is right. Explain how
to show m4  30.
C
B
2 3
1
D
4
A
F
E
Using the substitution property, you know that
m1 mBFD  150. m1  mBFD  mAFD by the
Angle Addition Postulate. The diagram shows that
mAFD  m4  180. Substitute 150° for mAFD to
show m4  30.
VERTICAL ANGLES THEOREM
EXAMPLE 6
Vertical angles are congruent.
Extra Example 6
Given: 1 and 2 are a linear pair, 2 and 3 are a
linear pair.
Prove: 1  3
Statements
1 2
3
Reasons
1. 1 and 2 are a linear pair 1. Given
2 and 3 are a linear pair
2. 1 and 2 are supplementary 2. Linear Pair Post.
2 and 3 are supplementary
3. 1  3
3.  Supplements Thm.
Checkpoint
1. Find the measures of the angles in the diagram given 1
and 2 are complementary and 1  3  4.
1
2
78°
3
4
m2  78, m1  m3  m4  12.
QUESTIONS?