Download Section 5.2 Proving That Lines are Parallel

Section 5.2
Proving That Lines are Parallel
Steven Shields and Will Swisher
Period 1
The Exterior Angle Inequality Theorem
• An exterior angle is formed when one side
of a triangle is extended.
Exterior
Angle
Theorem 30
• The measure of an exterior angle of a
triangle is greater than the measure of
either remote interior angle.
Exterior
Angle
Remote Interior
Angles
Theorem 30-Sample Problem
• Write a valid inequality and find the restrictions
on x.
50 < 2x-20 < 180
50+20 < 2x < 180+20
70 < 2x < 200
70/2 < x < 200/2
35 < x < 100
50
2x-20
Identifying Parallel Lines
• When two lines are cut by a transversal,
eight angles are formed. By proving
certain angles congruent, you can prove
lines II.
1 2
3
3
4
5
7
6
8
Theorem 31
• If two lines are cut by a transversal such
that two alternate interior angles are
congruent, the lines are II. (Alt. int. <s
congruent => II lines)
a
If <3 congruent
<4, then a II b
b
3
4
Theorem 31-Sample Problem
a
a
Is a II b?
b
5x
b
25
2x+15
If these lines are II, the alt. int. angles would be congruent.
5x=25
5(5)=25
x=5
2(5)+15=25
Yes, they are II because the alt. int. <s both equal 25.
Theorem 32
• If two lines are cut by a transversal such
that two alternate exterior angles are
congruent, the lines are II. (alt. ext. <s
congruent => II lines)
a
If <1 congruent
<2, then a II b
b
1
2
Theorem 32-Sample Problem
yx
x+20
Is x II y?
y
4x
52
x + 20 + 4x = 180 (These angles are suppl.)
5x + 20 = 180
x + 20 = 52
5x = 160
(32) + 20 = 52
x = 32
52 = 52
Therefore, the lines are parallel because alt. ext. <s congruent => II lines
Theorem 33
• If two lines are cut by a transversal such
that two corresponding angles are
congruent, the lines are II. ( corr. <s
congruent => II lines)
m
If <1 congruent
<2, then m II n
n
1
2
Theorem 33-Sample Problem
R
Q
P
3
4
T
S
If <3 congruent <4, then which lines are II? Write the theorem to
prove your answer.
QT II RS with transversal PS because Corr. <s congruent => II
lines.
Theorem 34
• If two lines are cut by a transversal such
that two interior angles on the same side
of the transversal are supplementary, the
lines are II.
If <1 suppl. <2,
then c II d.
c
1
d
2
Theorem 34-Sample Problem
If x=10, is w II z?
Explain.
w
Yes they are parallel
because one angle
would be 80 and the
other 100, so they
would be suppl.
Therefore the lines are
II by theorem 34.
z
8x
12x-20
Theorem 35
• If two lines are cut by a transversal such
that two exterior angles on the same side
of the transversal are supplementary, the
lines are II.
If <1 suppl. <2,
then a II b.
1
2
Theorem 35-Sample Problem
Is a II b ?
a
b
6x + 60 + 2x = 180
8x = 120
6x+60
2x
5x
6(15) + 60 + 5(15) = 180
225 = 180
X = 15
Therefore a is not II to b because the same side ext <s do
not add up to 180.
Theorem 36
• If two coplanar lines are perpendicular to
a third line, they are parallel.
a
b
c
a II b
Practice Problems
Name the theorem that proves a II b.
1.
2.
a
a
b
b
3.
a
b
80
100
Practice Problems Cont.
A
4.
B
1
D
Given: <1 congruent <2
Prove: BD II CE
C
2
E
Practice Problems Cont.
5.
Find the restrictions on x.
___< x < ___
x
125
Answers
1. Corr. <s congruent => II lines.
2. Alt. ext. <s congruent => II
lines.
3. Same side int. <s suppl. => II
lines.
4. Statements
Reasons
1. <1 congruent <2
1. Given
2. BD II CE
2. Corr. <s congruent => II lines
5. 0 < x < 125
Work Cited
• Rhoad, Richard, George Milauskas, Robert Whipple. Geometry for
Enjoyment and Challenge. Boston: McDougal Littell, 1997.