Download Lesson 3.1 Properties of Parallel Lines

3.1: Properties
of Parallel Lines
Every man
man dies,
dies, not
not every
every man
man really
really lives.
lives.
Every
-William Wallace
Identifying Angles
Transversal: A line that intersects two coplanar lines at two distinct
points.
l
a
b
m
k
c
How many angles are formed by a transversal?
Identifying Angles
Alternate Interior Angles: Nonadjacent interior angles
that lie on opposite sides of the transversal.
Same-Side Interior Angles: Angles that lie on the same
side of the transversal between the two lines it intersects
Corresponding Angles: Angles that lie on the same side
of the transversal in corresponding positions relative to
the two lines it intersects
Identifying Angles
5
1
6
Alternate Interior Angles:
1 and 2 are alternate interior angles
3
Also:
4
2
7
8

Same-Side Interior Angles:
 1 and 4 are same-side interior angles
(AKA co-interior angles)
Also:

 Angles:
Corresponding
1 and 7 are corresponding angles
Also:
Properties of Parallel Lines
t
1
l
Note: Notation for
parallel lines
2
m
Postulate 3-1: Corresponding Angles Postulate:
If a transversal intersects two parallel lines, then
corresponding angles are congruent.
1  2
Properties of Parallel Lines
Let’s say this
angle is 72°…
Alternate Interior Angles are congruent!!!
Properties of Parallel Lines
t
a
b
3 2
1
Theorem 3-1: Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate
interior angles are congruent.
1  3
Proof of Alternate Interior Angles Theorem
t
a
Given : a || b
Prove : 1  3
Statements
4
3 2
1
b
Reasons
a || b
2. 1  4
1.
3. 4  3
3.
4. 1  3
4.
1.
2.
Properties of Parallel Lines
Same-Side Interior Angles are supplementary!!!
Properties of Parallel Lines
t
a
b
3 2
1
Theorem 3-2: Same-Side Interior Angles Theorem
If a transversal intersects two parallel lines, then same-side
interior angles are supplementary.
m1 m2  180
Identifying Angles
Alternate Exterior Angles: Nonadjacent exterior angles
that lie on opposite sides of the transversal.
Same-Side Exterior Angles: Angles that lie on the same
side of the transversal outside of the two lines it intersects
Identifying Angles
5
1
6
4
2
7
8
Alternate Exterior Angles:
5 and 8 are alternate exterior angles
Also:
3

 Same-Side Exterior Angles:
5 and 7 are same-side exterior angles
(AKA co-exterior angles)
Also:


Properties of Parallel Lines
Alternate Exterior Angles are congruent!!!
Properties of Parallel Lines
a
1 2
b
3
Theorem 3-3: Alternate Exterior Angles Theorem
If a transversal intersects two parallel lines, then alternate
exterior angles are congruent.
1  3
Proof of Alternate Exterior Angles Theorem
a
Given : a || b
Prove : 3  1
Statements
1 2
4
b
3
Reasons
1.
1.
2. 3  4
2.
3. 4  1
3.
4. 3  1
4.
Properties of Parallel Lines
Same-Side Exterior Angles are supplementary!!!
Properties of Parallel Lines
a
1 2
b
3
Theorem 3-4: Same-Side Exterior Angles Theorem
If a transversal intersects two parallel lines, then same-side
exterior angles are supplementary.
m2  m3  180
Let’s Apply What We Have Learned, K?
Find the values of x and y in the diagram below.
50°
x° y°
70°
Let’s Apply What We Have Learned, K?
Find the values of x and y in the diagram below.
52°
y°
66°
x°
3.1: Properties
of Parallel Lines
HOMEWORK: 3.1: #5-9, 11-16, 19-25
TERMS: transversal, alternate interior (exterior) angles, sameside interior (exterior) angles, corresponding angles
Every man dies, not every man really lives.
-William Wallace