Download Symbol: ⊥

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.
A
D
X
C
B
Given: AB ^ CD
Possible Conclusions:
DXB is a right angle. Definition of Perpendicular Lines.
CXB is a right angle. Definition of Perpendicular Lines.
CXA is a right angle. Definition of Perpendicular Lines.
AXD is a right angle. Definition of Perpendicular Lines.
Once we have said one of these, then we can say…
mAXD = 90
Definition of a right angle.
A
D
X
C
B
Given: AXD is a right angle
Possible Conclusions:
mAXD = 90 Definition of a Right Angle
AB ^ DC Definition of a Perpendicular Lines
Theorem: If two lines are perpendicular, then
they form congruent adjacent angles.
Given: Two lines are perpendicular.
Prove: The lines form congruent adjacent angles.
A
D
X
C
Given: AB ^ DC
B
Prove: AXD @ DXB
D
Given: AB ^ DC
A
Prove: AXD @ DXB
Statements
1. AB ^ DC
X
Reasons
C
1. Given
2. AXD is a right angle. 2. Definition of
DXB is a right angle. Perpendicular Lines
3. mAXD = 90
mDXB = 90
3. Definition of a
right angle.
4. mAXD = mDXB
AXD @ DXB
4. Substitution
B
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!
A
D
X
C
B
Given: AXD @ DXB
Prove: AB ^ DC
D
Given: AXD @ DXB
Prove: AB ^ DC
Statements
1. mAXD = mDXB
AXD @ DXB
A
X
Reasons
1. Given
B
C
2. mAXD + mDXB = 180 2. Angle Addition Postulate
3. mAXD + mAXD = 180 3. Substitution
2mAXD = 180
4. mAXD = 90
4. Division Property
5. AXD is a right angle.
5. Definition of a right angle.
6. AB ^ DC
6. Definition of
perpendicular Lines
Theorem: If the exterior sides of two
adjacent acute angles are perpendicular,
then the angles are complementary.
Given: OA ^ OC
Prove: AOB and BOC are complementary
angles.
A
B
O
C
Given: OA ^ OC
A
Prove: AOB and BOC are complementary angles.
Statements
Reasons
O
B
C
1. OA ^ OC
1. Given
2. AOC is a right angle.
2. Definition of
Perpendicular Lines
3. Definition of a right angle.
3. mAOC = 90
4. mAOB + mBOC = mAOC 4. Angle Addition Postulate
5. mAOB + mBOC = 90 5. Substitution
6. AOB and BOC are
complementary angles
6. Definition of Complementary
Angles
Check for Understanding
Pg.57 CE #6-11