1. Section 2-5: Perpendicular Lines & Proofs
Perpendicular Lines – two lines that intersect to form right angles.
Symbol: ^
Review: You can write all definitions as biconditional statements…
Biconditional: Two lines are perpendicular, if and only if, they intersect to form right angles.

2. D A B C Possible Conclusions: Given: AB ^ CD X ÐDXB is a right angle.
Definition of Perpendicular Lines.
ÐCXB is a right angle.
Definition of Perpendicular Lines.
ÐCXA is a right angle.
Definition of Perpendicular Lines.
ÐAXD is a right angle.
Definition of Perpendicular Lines.
Once we have said one of these, then we can say…
mÐAXD = 90
Definition of a right angle.

3. D A B C Possible Conclusions: mÐAXD = 90 AB ^ DC
Given: ÐAXD is a right angle A B X C
Possible Conclusions:
mÐAXD = 90
Definition of a Right Angle
AB ^ DC
Definition of a Perpendicular Lines

4. D A B C Given: AB ^ DC Prove: ÐAXD @ ÐDXB
Theorem: If two lines are perpendicular, then they form congruent adjacent angles.
Given:
Two lines are perpendicular.
Prove:
The lines form congruent adjacent angles. D
Given: AB ^ DC
Prove: ÐDXB A B X C

5. D A B C X Given: AB ^ DC Prove: ÐAXD @ ÐDXB Statements Reasons
2. ÐAXD is a right angle.
ÐDXB is a right angle.
2. Definition of
Perpendicular Lines
3. mÐAXD = 90
mÐDXB = 90
3. Definition of a
right angle.
4. mÐAXD = mÐDXB
ÐDXB
4. Substitution

6. D A B C Given: ÐAXD @ ÐDXB Prove: AB ^ DC
Theorem: If two lines form congruent adjacent angles, then the lines are perpendicular.
What is the relationship between this theorem and the last one?
They are converses! D A
Given: ÐDXB
Prove: AB ^ DC B X C

7. D A B C X Given: ÐAXD @ ÐDXB Prove: AB ^ DC Statements Reasons
1. mÐAXD = mÐDXB
ÐDXB
1. Given
2. mÐAXD + mÐDXB = 180
2. Angle Addition Postulate
3. mÐAXD + mÐAXD = 180
2mÐAXD = 180
3. Substitution
4. mÐAXD = 90
4. Division Property
5. ÐAXD is a right angle.
5. Definition of a right angle.
6. Definition of perpendicular Lines
6. AB ^ DC

8. Theorem: If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Given: OA ^ OC
Prove: ÐAOB and ÐBOC are complementary angles. A B O C

9. 4. Angle Addition Postulate
Given: OA ^ OC
Prove: ÐAOB and ÐBOC are complementary angles. A B O C
Statements
Reasons
1. OA ^ OC
1. Given
2. ÐAOC is a right angle.
2. Definition of Perpendicular Lines
3. mÐAOC = 90
3. Definition of a right angle.
4. Angle Addition Postulate
4. mÐAOB + mÐBOC = mÐAOC
5. mÐAOB + mÐBOC = 90
5. Substitution
6. ÐAOB and ÐBOC are
complementary angles
6. Definition of Complementary Angles

10. 2. Ð1 and Ð2 are complementary angles
Given: AO ^ CO
Prove: Ð1 and Ð3 are complementary angles 2 O 1 C 3
Statements
Reasons
1. Given
1. AO ^ CO
2. If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.
2. Ð1 and Ð2 are complementary angles
3. Definition of Complementary Angles
3. mÐ1 + mÐ2 = 90
4. Ð3 or mÐ2 = mÐ3
4. Vertical Angle Theorem
5. mÐ1 + mÐ3 = 90
5. Substitution
6. Ð1 and Ð3 are complementary angles
6. Definition of Complementary Angles