Section 2-5

Perpendicular Lines

& Proofs

Perpendicular Lines – two lines that intersect to form right angles.

Biconditional: Two lines are perpendicular, if and only if, they

intersect to form right angles.

Symbol:

A B

C

DX

Possible Conclusions:

DXB is a right angle.CXB is a right angle.CXA is a right angle.AXD is a right angle.

Once we have said one of these, then we can say…

mAXD = 90 Definition of a right angle.

Given: AB CD

A B

C

DX

Possible Conclusions:

Given: AXD is a right angle

mAXD = 90 Definition of a Right Angle

Definition of a Perpendicular Lines

AB DC

Theorem 2-4: If two lines are perpendicular, then they form congruent

adjacent angles.Given:

Prove:

Two lines are perpendicular.

The lines form congruent adjacent angles.

A B

C

DX

Given: AB DC

Prove: AXD DXB

Given: AB DC

Prove: AXD DXBA B

C

DX

Statements Reasons

1. AB DC 1. Given

4. mAXD = mDXBAXD DXB

2. AXD is a right angle. DXB is a right angle.

2. Definition of Perpendicular Lines

3. mAXD = 90 mDXB = 90

3. Definition of a right angle.

4. Substitution

PROOF OF THEOREM 2-4:

Theorem 2-5: 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!

A B

C

DX Given: AXD

DXB

Prove: AB DC

Given: AXD DXB

Prove: AB DCA B

C

DX

Statements Reasons

6. AB DC

1. Given1. AXD DXB mAXD = mDXB

5. AXD is a right angle.

2. Angle Addition Postulate2. mAXD + mDXB = 180

3. Substitution

6. Definition of perpendicular Lines

3. mAXD + mAXD = 180 2mAXD = 180

4. mAXD = 90 4. Division Property5. Definition of a right angle.

PROOF OF THEOREM 2-5:

Theorem 2-6: If the exterior sides of two adjacent acute angles are perpendicular, then the angles

are complementary.

A

O C

B

Given: OA OC

Prove: AOB and BOC are complementary

angles.

Given: OA OC

Prove: AOB and BOC are complementary angles.

A

O C

B

Statements Reasons

6. AOB and BOC are complementary angles

1. Given1. OA OC

2. AOC is a right angle.

6. Definition of Complementary Angles

4. mAOB + mBOC = mAOC 4. AngleAddition Postulate

2. Definition of Perpendicular Lines

5. mAOB + mBOC = 90

3. mAOC = 90

5. Substitution

3. Definition of a right angle.

PROOF OF THEOREM 2-6:

Given: AO COProve: 1 and 3 are complementary angles

A

C

O 12

3Statements Reasons1. AO CO

2. 1 and 2 are complementary angles

4. 2 3,

6. 1 and 3 are complementary angles

1. Given2. If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.

4. Vertical Angle Theorem

6. Definition of Complementary Angles

3. m1 + m2 = 90

5. m1 + m3 = 90

3. Definition of Complementary Angles

5. Substitution

EXAMPLE 4:THEOREM 2-6

m2 = m3