Download Perpendicular Lines & Proofs

Section 2-5
Perpendicular Lines
& Proofs
Perpendicular Lines – two lines that
intersect to form right angles.
Symbol: ^
Biconditional: Two lines are
perpendicular, if and only if, they
intersect to form right angles.
A
D
X
Given: AB ^ CD
B
C
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…
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 2-4: 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
PROOF OF THEOREM 2-4:
A
Given: AB ^ DC
Prove: AXD @ DXB
Statements
1. AB ^ DC
X
Reasons
1.
C
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 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
D
X
C
B
Given: AXD @ DXB
Prove: AB ^ DC
D
PROOF OF THEOREM 2-5:
Given: AXD @ DXB
A
Prove: AB ^ DC
Statements
1. AXD @ DXB
mAXD = mDXB
1.
Reasons
Given
X
B
C
2. mAXD + mDXB = 180 2. Angle Addition Postulate
3. mAXD + mAXD = 180 3.
2mAXD = 180
Substitution
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 2-6: 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
PROOF OF THEOREM 2-6:
A
Given: OA ^ OC
B
Prove: AOB and BOC are complementary angles.
Statements
Reasons
O
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
EXAMPLE 4:
Given: AO ^ CO
Prove: 1 and 3 are
complementary angles
Statements
1. AO ^ CO
2. 1 and 2 are
complementary angles
3.
m1 + m2 = 90
4. 2 @ 3, m2 = m3
5.
m1 + m3 = 90
6. 1 and 3 are
complementary angles
A
O
3
1.
2
THEOREM
2-6
1
C
Reasons
Given
2. If the exterior sides of two
adjacent acute angles are
perpendicular, then the angles are
complementary.
3. Definition of Complementary
Angles
4. Vertical Angle Theorem
5.
Substitution
6. Definition of Complementary
Angles