Transcript 3.1 Properties of Parallel Lines

Properties of Parallel Lines
Properties of Parallel Lines

Transversal: line that intersects two coplanar
lines at two distinct points
Transversal
Properties of Parallel Lines
The pairs of angles formed have special names…
t
Transversal
5
1
4
7
2
8
6
3
l
m
Alternate Interior Angles
<1 and <2
<3 and <4
t
5
1
4
7
2
8
6
3
l
m
Same-side Interior Angles
<1 and <4
<2 and <3
t
5
1
4
7
2
8
6
3
l
m
Corresponding Angles
<1 and <7
<2 and <6
<3 and <8
<4 and <5
t
5
1
4
7
2
8
6
3
l
m
Properties of Parallel Lines
Postulate 3-1
Corresponding Angles Postulate
If a transversal intersects two parallel lines, then
corresponding angles are congruent
t
line l || line m
1
l
2
m
m<1 =m <2
Properties of Parallel Lines
Theorem 3-1
Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate
interior angles are congruent.
t
line l || line m
l
3
1
2
m
m<2 = m<3
Properties of Parallel Lines
Theorem 3-2
Same-Side Interior Angles Theorem
If a transversal intersects two parallel lines, then same-side
interior angles are supplementary.
t
line l || line m
3
l
1
2
m
m<1 + m<2 = 180
Two-Column Proof
Given: a || b
Prove: m<1 = m<3
4
a
3
b
Statements
Reasons
1.
1. Given
a || b
3.
1   4
3   4
4.
3  1
2.
t
1
2. Corr. Angle Postulate
3. Vert. Angles
4. Set Statement 1 = Statement 2
* This proves why alternate interior angles are congruent *
Two-Column Proof
Given: a || b
Prove: <1 and <2 are
supplementary
Statements
1. a || b
1  3
3. 2 is supp. to 3
2.
t
3
a
1
b
2
Reasons
1. Given
2. Corr. Angle Postulate
3. Consecutive Angles
Finding Angle Measures
<1
<2
<3
<4
<5
<6
<7
<8
c
d
8
7
a || b
c || d
a
6
2
50°
5
4
1
3
b
Using Algebra to Find Angle Measures
Find the value of x and y.
x=
y=
50°
x
y
70°
▲
2x
y
▲
(y – 50)