1. COMPARING TYPES OF PROOFS
Three types of proofs:
TWO-COLUMN PROOF is the most formal type. Lists numbered statements in the left column and the reason for each statement in the right column.
PARAGRAPH PROOF describes the logical argument with sentences. It is more conversational than a two-column proof.
FLOW PROOF uses the same statements and reasons as a two-column proof, but the logical flow connecting the statements is indicated by arrows.
3.2
Proof and Perpendicular Lines

2. 5  7 Congruent Supplement Theorem
Two-Column Proof 5 6 7
GIVEN
5 and 6 are a linear pair.
6 and 7 are a linear pair.
PROVE
Statements
Reasons 1
5 and 6 are a linear pair. Given
6 and 7 are a linear pair.
5 and 6 are supplementary. Linear Pair Postulate
6 and 7 are supplementary. 2 3
5  Congruent Supplement Theorem
3.2
Proof and Perpendicular Lines

3. 5 7 3.2 5 and 6 are a linear pair. 6 and 7 are a linear pair.
Paragraph Proof 5 6 7
GIVEN
5 and 6 are a linear pair.
6 and 7 are a linear pair.
PROVE
Because 5 and 6 are a linear pair, the Linear Pair Postulate says that 5 and 6 are supplementary. The same reasoning shows that 6 and 7 are supplementary. Because 5 and 7 are both supplementary to 6, the Congruent Supplements Theorem says that 5  7.
3.2
Proof and Perpendicular Lines

4. 5 7 5  7 3.2 5 and 6 are a linear pair.
Flow Proof 5 6 7
GIVEN
5 and 6 are a linear pair.
6 and 7 are a linear pair.
PROVE
5 and 6 are a linear pair.
5 and 6 are supplementary.
5  7
Given
Linear Pair Postulate
Congruent
Supplements
Theorem
6 and 7 are a linear pair.
6 and 7 are supplementary.
Given
Linear Pair Postulate
3.2
Proof and Perpendicular Lines

5. • B Finding Angle Measures
Solve for x and y. Then find the angle measure.
( x + 15)˚
( 3x + 5)˚
( y + 20)˚
( 4y – 15)˚ D • C
• B
A • E
Use the fact that the sum of the measures of angles that form a linear pair is 180˚.
SOLUTION
Use substitution to find the angle measures (x = 40, y = 35).
m AED = ( 3 x + 15)˚ = (3 • )˚
= 125˚
m AED + m DEB = 180°
m AEC + mCEB = 180°
m DEB = ( x + 15)˚ = ( )˚
= 55˚
( 3x + 5)˚ + ( x + 15)˚ = 180°
( y + 20)˚ + ( 4y – 15)˚ = 180°
m AEC = ( y + 20)˚ = ( )˚
= 55˚
4x + 20 = 180
5y + 5 = 180
m CEB = ( 4 y – 15)˚ = (4 • 35 – 15)˚
4x = 160
= 125˚
5y = 175
So, the angle measures are 125˚, 55˚, 55˚, and 125˚. Because the vertical angles are congruent, the result is reasonable.
x = 40
y = 35

6. m AEC = m DEB = 48˚ • • B 6x˚ = (4x + 16)˚ 6x˚ = (4x + 16)˚
Finding Angle Measures
Solve for x and y. Then find the angle measure. D •
A • E
6x˚
( 4x + 16)˚ • C
11y˚
• B
SOLUTION
 AEC and  DEB are vertical angles.
m AEC and m CEB are a linear pair.
m AEC = m DEB
6x˚ = (4x + 16)˚
m AEC + m CEB = 180 ˚
6x˚ = (4x + 16)˚
m CEB = 11y˚= 132 ˚
6(8)˚ = (4(8) + 16)˚
6x˚ + 11y˚= 180 ˚
2x = 16
m CEB = m AED = 132 ˚
48˚ = ( )˚
6(8)˚ + 11y˚= 180 ˚
x = 8
48˚ = 48˚
48˚ + 11y˚= 180 ˚
11y˚= 132 ˚
m AEC = m DEB = 48˚
y˚= 12 ˚
Review