Theorems About Perpendicular Lines

Comparing Types of Proofs

• Two-Column Proof• Paragraph Proof• Flow Proof– Uses arrows to show the flow of the logical

argument

Right Angle Congruence Theorem

1. Two Column Proof:

All right angles are congruent.Given: 1 and 2 are right anglesProve: 1 ≅ 2

Statement Reason

1 and 2 are right angles Given

m 1 = 90o, m 2 = 90o Definition of a right angle

m 1 = m 2 Transitive property of equality

1 ≅ 2 Definition of congruent angles

Because angles 1 and 2 are right angles, their measures are equal to 90o, by the definition of right angles. Hence by the transitive property of equality, the measure of angle 1 is equal to the measure of angle 2. By the definition of congruent angles, angle 1 is congruent to angle 2.

2. Paragraph Proof

3. Flow Proof

Theorem 3.1

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

Theorem 3.2

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

Theorem 3.3

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

Proof of Theorem 3.2

Statement ReasonAB BC Given ABC is a right angle Definition of perpendicular lines

m ABC = 90o Definition of a right anglem 1 + m 2 = m ABC Angle addition postulatem 1 + m 2 = 90o Substitution property of

equality 1 + 2 are complementary Definition of complementary

angles

Prove: 1 + 2 are complementary

Practice and Homework

• Workbook: Exercise 3.2• Textbook: Exercise 3.2 p138: 1-27, odd