3-1 PROPERTIES OF PARALLEL LINES (p. 115-121)
A transversal is a line that intersects two other coplanar lines at two distinct points.
Example: Sketch a transversal to two non-parallel lines. Label the eight angles that are
formed with the numbers 1-8.
Pairs of the eight angles have names that are suggested by their positions. Knowing the
definition of a few terms will be helpful.
Interior means situated inside the two lines being cut by the transversal.
Exterior means situated outside the two lines being cut by the transversal.
Alternate means situated on different sides of the transversal.
Same-side means situated on the same side of the transversal.
Corresponding means situated in the same position with respect to the intersections of the
two lines and the transversal.
Example: Using the above diagram, identify alternate interior angles, same-side interior
angles, and corresponding angles. Also, briefly mention alternate exterior angles and
same-side exterior angles.
Example:
C
A
4 1
3 2
D
E
5
8
7
6
B
F
Which angle forms a pair of same-side interior angles with 2 ?
Which angle forms a pair of corresponding angles with 8 ?
Which angle forms a pair of alternate interior angles with 7 ?
Is there an angle that forms a pair of alternate interior angles with 3?
Use the next diagram for the following postulate and two theorems. The two horizontal
lines are parallel.
E
1
3
5
A
7
2
4
6
8
B
D
Postulate 3-1 Corresponding Angles Postulate
If a transversal intersects two parallel lines, then corresponding angles are
congruent.
Short form:
lines  corr. s 
Example: Use the above diagram and list the congruent corresponding angle pairs.
Theorem 3-1 Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are
congruent.
Short form:
lines  alt. int. s 
Example: Use the above diagram and list the congruent alternate interior angle pairs.
Theorem 3-2 Same-side Interior Angles Theorem
If a transversal intersects two parallel lines, then the same-side interior angles are
supplementary.
Short form:
lines  same side int. s supp.
Example: Use the above diagram and list the supplementary same-side interior angle
pairs.
We will now prove Theorem 3-1 using a new proof format called the two-column proof.
See p. 117 for the format. The left column represents the statements and the right column
represents the reasons.
For the proof of Theorem 3-1, notice the new symbol for the phrase “is parallel to.”
Also, note the arrowhead tick marks which indicate parallel lines. To prove this theorem,
we use concepts that we learned prior to being introduced to this new theorem. These
“old” concepts make up reasons 2-4.
Before you actually write a complete proof, you might want to first write a plan for the
proof. This is comparable to writing a short outline for a longer essay or paper.
Example: After studying the plan in Example 3 on p. 117, write a two-column proof.
When you have two parallel lines and a transversal, and you know the measure of one
angle, you can determine the measures of the seven remaining angles. To do this, apply
some of our newer theorems and a postulate.
Example: In the following diagram, two parallel lines are being cut by a transversal.
Find the measures of the seven remaining angles. Which theorem or postulate justifies
each answer?
E
63
A
B
D
Example:
C
x
58
y
z
44
A
B
In the above diagram, the two horizontal lines are parallel. Find the values of x, y, and z.
Do 5 on p. 118.
Homework p. 118-121: 2,4,7,10,13,15,16,18,25,27,30,38,40,44,46,50
18. One. All the rest are congruent or supplementary to this one.
x  y  180
25. Solve this system: 
x  3y