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Chapter 2
Section 5
Perpendicular lines
Define: Perpendicular lines (^)
o
• Two lines that intersect to form right or 90 angles
The box shows you the right angle
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
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
t
Given: r ^ t
2 1
Prove: <1 @ <2
Hypothesis
conclusion
None
r^t
then m<1= 90 & m<2 = 90
o
If m<1= 90 & m<2 = 90
If m<1= m<2
reason
given
o
If r ^ t
r
o
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.
2
1
Angles 1 and 2 are a linear pair.
Perpendicular Line Theorems
• If two lines form congruent adjacent angles, then they are
perpendicular
t
Given: R1 @ R2
2 1
Prove: r ^ t
Hypothesis
conclusion
None
R1 @ R2 or mR1 = mR2
reason
If R1 and R<2 are a linear pair
then mR1 + mR2 = 180
If mR1 = mR2 and
then 2 (mR1) = 180
mR1 + mR2 = 180
If 2 (mR1) = 180
If mR1 = 90
o
o
r
o
o
given
Angle addition post
substitution
o
then mR1 = 90
then r ^ t
o
Division property
Def of ^ lines
Define: Perpendicular Pair of Angles
• Two adjacent acute angles whose exterior
sides are perpendicular.
2
1
Angles 1 and 2 are a perpendicular pair.
Perpendicular Pair Theorem
• If two angles are a perpendicular pair, then
the angles are complementary
Given: R1 and R2 are a perpendicular pair
A
X
Prove: R1 and R2 are complementary angles
2
B
1
C
Proof of ^ Pair Theorem
A
Given: R1 and R2 are a perpendicular pair
X
Prove: R1 and R2 are complementary angles
2
B
Hypothesis
conclusion
None
R1 and R2 are a ^ pair
then BA ^ BC
If BA ^ BC
then mRABC = 90
If X is in the interior of RABC
then mRABC = mR1 +mR2
If mRABC = mR1 +mR2 and
then mR1 +mR2 = 90
If mR1 +mR2 = 90
o
C
reason
If R1 and R2 are a ^ pair
mRABC = 90
1
given
Def of ^ pair
o
Def of ^ lines
o
o
then R1 & R2 are comp. R<‘s
Angle addition postulate
Transitive property or
substitution
Def of comp. R‘s
summary
Definition of ^
^
90 or right
^
^ line theorems
2 lines forming @ adjacent R’s
^ pair theorem^ pair
complementary R’s
Practice work
• P 58 we 1-25all