Chapter 2

Section 5

Perpendicular lines

Define: Perpendicular lines ()

• Two lines that intersect to form right or 90o angles

Remember all definitions are Biconditionals:

If two lines are perpendicular then they form right angles

If two lines form right angles then they are perpendicular

The box shows you the right angle

Perpendicular Line Theorems

• If two lines are perpendicular, then they form congruent adjacent angles

• If two lines form congruent adjacent angles, then they are perpendicular

Perpendicular Line Theorems• If two lines are perpendicular, then they form congruent adjacent

angles

12r

t

Given: r t

Prove: <1 <2

Hypothesis conclusion reason

None r t given

If r t

If m<1= 90o & m<2 = 90

o

If m<1= m<2

then m<1= 90o & m<2 = 90

o def of perpendicular lines

then m<1 = m<2 transitive prop / substitution

then <1 <2 def of congruent angles

Define: Linear Pair of Angles

• Two adjacent angles whose exterior sides are opposite rays.

12

Angles 1 and 2 are a linear pair.

Perpendicular Line Theorems• If two lines form congruent adjacent angles, then they are

perpendicular

12r

t

Given: 1 2

Prove: r t

Hypothesis conclusion reason

None 1 2 or m1 = m2 given

If 1 and <2 are a linear pair

If m1 = m2 and

m1 + m2 = 180o

If 2 (m1) = 180o

then m1 + m2 = 180o Angle addition post

then 2 (m1) = 180o substitution

then m1 = 90o Division property

If m1 = 90o

then r t Def of lines

Define: Perpendicular Pair of Angles

• Two adjacent acute angles whose exterior sides are perpendicular.

12

Angles 1 and 2 are a perpendicular pair.

Perpendicular Pair Theorem

• If two angles are a perpendicular pair, then the angles are complementary

12

A

B C

XGiven: 1 and 2 are a perpendicular pair

Prove: 1 and 2 are complementary angles

Proof of Pair Theorem

12

A

B C

XGiven: 1 and 2 are a perpendicular pair

Prove: 1 and 2 are complementary angles

Hypothesis conclusion reason

None 1 and 2 are a pair given

If 1 and 2 are a pair

If X is in the interior of ABC

Def of pair

then mABC = 90o Def of lines

then mABC = m1 +m2 Angle addition postulate

If mABC = m1 +m2 and

mABC = 90o

then m1 +m2 = 90o Transitive property or

substitution

then BA BC

If BA BC

If m1 +m2 = 90o

then 1 & 2 are comp. <‘s Def of comp. ‘s

Definition of 90 or right

line theorems

2 lines forming adjacent ’s

pair theorem-

pair complementary ’s

summary

Practice work

• P 58 we 1-25all