GOAL 1 COMPARING TYPES OF PROOFS

EXAMPLE 1Vocabulary

•two-column proof

•paragraph proof

•flow proof

3.2PROOF AND PERPENDICULAR LINES

Be sure you identify the similarities and differences in the

three types of proof.

Note that the statements and reasons are the same, but are

written in a different form.

Extra Example 1Given: AB = CD

Prove: AC = BDA B C D

Statements Reasons

1. AB = CD 1. Given

2. 2.

3. 3.

4. 4.

AB + BC = BC + CD Addition prop. of =

AB + BC = AC and BC + CD = BD

Segment Addition post.

AC = BD Substitution prop. of =

Now use your two-column proof to write a paragraph proof.

Extra Example 1 Paragraph ProofGiven: AB = CD

Prove: AC = BDA B C D

Because AC = BD, the addition property of equality says that AB + BC = BC + CD. Since AB + BC = AC andBC + CD = BD by the segment addition post., it follows from the substitution prop. of equality that AC = BD.

Now write a flow proof.

Extra Example 1 Flow ProofGiven: AB = CD

Prove: AC = BDA B C D

AB = CD AB + BC = BC + CD

AB + BC = AC, BC + CD = BD

AC = BD

Given Addition prop. of =

Segment Addition post.

Substitution prop. of =

GOAL 2 PROVING RESULTS ABOUT PERPENDICULAR LINES

3.2PROOF AND PERPENDICULAR LINES

EXAMPLE 2

3.1 If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.

3.2 If two sides of two adjacent angles are perpendicular, then the angles are complementary.

3.3 If two lines are perpendicular, then they intersect to form four right angles.

THEOREMS ABOUT PERPENDICULAR LINES

Extra Example 2Write a two-column proof of Theorem 3.2.

Given:

Prove:

BA BC����������������������������

1 and 2 are complementary.

A

B C12

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

BA BC����������������������������

Given

is a right angle.ABC Def. of lines

90m ABC Def. of right s

1 2m m m ABC Addition Post.

1 2 90m m Subs. Prop. of =

1 and 2 are complementary. Def. of comp. s

CheckpointUse the following to write a paragraph proof of the Congruent Supplements Theorem.

Given: Prove: 2 4 1 is supplementary to 2. 3 is supplementary to 4.

1 3

1 2 3 4

A solution is on the next slide.

Checkpoint SolutionSince is supplementary to and is supplementary to , and by the definition of supplementary angles. Therefore the symmetric and transitive properties of equality say that It is given that so by the definition of congruent angles, The subtraction property of equality allows the statement Finally, by the definition of congruent angles,

1 2 34 1 2 180m m 3 4 180m m

1 2m m 3 4.m m 2 4,

2 4.m m 1 3.m m 1 3.

QUESTIONS?