2.6 PROVING STATEMENTS ABOUT ANGLES

GOAL 1 Justify statements about congruent angles.

GOAL 2 Prove properties about special pairs of angles

What you should learn

Properties of special pairs of angles help you determine angles in real-life applications, such as design work.

Why you should learn it

GOAL 1 PROPERTIES OF CONGRUENT ANGLES

VOCABULARY

EXAMPLE 1

2.6 PROVING STATEMENTS ABOUT ANGLES

PROPERTIES OF ANGLE CONGRUENCE

ReflexiveSymmetricTransitive

Extra Example 1

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

Given:

Prove:

1 2, 3 4, 2 3

1 4

A C

B

12 34

1 2, 2 3

3 4

Given

1 3 Transitive Prop. of

Given

1 4 Transitive Prop. of

EXAMPLE 2

Extra Example 2

Given: Prove: 1 63 , 1 3, 3 4m 4 63m

1 4 Transitive Prop. of

1 4m m Def. of s

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

1 3, 3 4

4 63m Subs. Prop. of =

1 2

43

1 63m Given

EXAMPLE 3

RIGHT ANGLECONGRUENCE THEOREM

All right angles are congruent.

Extra Example 3Given:

Prove:

, are right angles; DAB ABC ABC BCD

DAB BCD

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

, are right anglesDAB ABC

DAB ABC Right s are

ABC BCD Given

DAB BCD Transitive Prop. of

A B

CD

CheckpointGiven:

Prove:

, are right angles, AFC BFD BFD CFE

AFC CFE

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

, are right anglesAFC BFD

AFC BFD Right s are

BFD CFE Given

AFC CFE Transitive Prop. of

A

BC

D

EF

GOAL 2 USING CONGRUENCE OF ANGLES

2.6 PROVING STATEMENTS ABOUT ANGLES

Two angles supplementary to the same angle (or congruent angles) are congruent

CONGRUENT SUPPLEMENTS THEOREM

In proofs, these may be abbreviated as Supp. Thm. and Comp. Thm.

CONGRUENT COMPLEMENTS THEOREM

Two angles complementary to the same angle (or congruent angles) are congruent

EXAMPLE 4

Extra Example 4

Statements Reasons1. 1. Given

2. 2. 3. 3. 4. 4.

1 24 , 3 24 ,1 and 2 are complementary,3 and 4 are complementary

m m

1 3m m Transitive Prop. of = 1 3 Def. of s 2 4 Complements Thm.

Given:

1 24 , 3 24 ,

1 and 2 are complementary,

3 and 4 are complementary

m m

2 4 Prove:

1 2 34

Checkpoint1. In a diagram, are supplementary and are supplementary. Explain how to show that

1 and 2 2 and3 1 3.

Using the definition of supplementary angles, So

by the transitive property ofequality. So by the subtraction property ofequality. Therefore, by the definition of congruent angles.

1 2 180 and 2 3 180 .m m m m 1 2 2 3m m m m

1 3m m 1 3

EXAMPLE 5

LINEAR PAIR POSTULATE

If two angles form a linear pair, then they are supplementary.

Extra Example 5

EXAMPLE 6

In the diagram is right. Explain how to show

1 60 and m BFD 4 30 .m

Using the substitution property, you know that by the

Angle Addition Postulate. The diagram shows that Substitute 150° for to

show

1 150 .m m BFD 1m m BFD m AFD

4 180 .m AFD m m AFD4 30 .m

A

BC

D

EF1

2 34

VERTICAL ANGLES THEOREM

Vertical angles are congruent.

Extra Example 6Given: are a linear pair, are a linear pair.

Prove:

1 and 2 2 and 3

1 3

Statements Reasons

1. 1. Given

2. 2.

3. 3.

1 and 2 are a linear pair 2 and 3 are a linear pair

1 and 2 are supplementary 2 and 3 are supplementary

Linear Pair Post.

1 3 Supplements Thm.

1 23

Checkpoint1. Find the measures of the angles in the diagram given and are complementary and

12 1 3 4.

2 78 , 1 3 4 12 .m m m m

1 278°

34

QUESTIONS?