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(3.1) Properties of Parallel Lines
What are we learning?
Students will…
1. Identify angles formed by two lines and a transversal.
2. Proving and using properties of parallel lines.
Evidence Outcome: Prove geometric theorems (lines, angles,
triangles, parallelograms).
Purpose (Relevancy): Do you think it is important for architects and
builders to know if lines are parallel or perpendicular?
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
Two-Column Proof
Proof of Alternate Interior Angles Theorem
t
a
Given : a || b
4
3 2
1
b
Prove : 1  3
Statements
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
1 2
4
b
Prove : 3  1
Statements
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?
Find the values of x and y in the diagram below. Use a two-column
proof to show your work. Under statements, write each step and
under reasons, write the definition, property, postulate, or theorem
that supports your ideas. The first statement should be the lines are
parallel and the first reason should be given.
50°
x°
y°
70°
Let’s Apply What We Have Learned, K?
Find the values of x and y in the diagram below. Use a two-column
proof to show your work. Under statements, write each step and
under reasons, write the definition, property, postulate, or theorem
that supports your ideas. The first statement should be the lines are
parallel and the first reason should be given.
52°
y°
66°
x°
HOMEWORK: (3.1) Pg. 131, #5-9 all, 11-16 all, 19-25 all
TERMS: transversal, alternate interior (exterior) angles, sameside interior (exterior) angles, corresponding angles
Thinking Page:
Using a ruler, draw two parallel lines, cut by a transversal. Use
the symbol for parallel and label two parallel lines c and d.
Label the transversal as line t.
1. Put a star to show one set of corresponding angles.
2. Put a checkmark to show one set of same-side
exterior angles.
3. Put a dot to show one set of vertical angles.
4. Put an arch to show alternate interior angles.