1. Parallels § 4.2 Parallel Lines and Transversals
§ 4.3 Transversals and Corresponding Angles
§ 4.4 Proving Lines Parallel

2. Parallel Lines and Planes
What You'll Learn
You will learn to describe relationships among lines, parts of lines, and planes.
In geometry, two lines in a plane that are always the same distance apart are ____________.
parallel lines
No two parallel lines intersect, no matter how far you extend them.

3. Parallel Lines and Transversals
What You'll Learn
You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel lines and a transversal.

4. Parallel Lines and Transversals
In geometry, a line, line segment, or ray that intersects two or more lines at
different points is called a __________
transversal B A l m 1 2 4 3 5 6 8 7
is an example of a transversal. It intercepts lines l and m.
Note all of the different angles formed at the points of intersection.

5. Parallel Lines and Transversals
Definition of
Transversal
In a plane, a line is a transversal iff it intersects two or more
Lines, each at a different point.
The lines cut by a transversal may or may not be parallel. l m 1 2 3 4 5 7 6 8
Parallel Lines
t is a transversal for l and m. t 1 2 3 4 5 7 6 8 b c
Nonparallel Lines
r is a transversal for b and c. r

6. Parallel Lines and Transversals
Two lines divide the plane into three regions.
The region between the lines is referred to as the interior.
The two regions not between the lines is referred to as the exterior.
Exterior
Interior

7. Parallel Lines and Transversals
When a transversal intersects two lines, _____ angles are formed.
eight
These angles are given special names. l m 1 2 3 4 5 7 6 8 t
Interior angles lie between the
two lines.
Exterior angles lie outside the
two lines.
Alternate Interior angles are on the
opposite sides of the transversal.
Alternate Exterior angles are
on the opposite sides of the
transversal.
Same Side Interior angles are on the same side of the transversal. ?

8. Parallel Lines and Transversals
Theorem 3-1
Alternate
Interior
Angles
If two parallel lines are cut by a transversal, then each pair of
Alternate interior angles is _________.
congruent 1 2 4 3 5 6 8 7

9. Parallel Lines and Transversals
Theorem 3-2
Same Side
Interior
Angles
If two parallel lines are cut by a transversal, then each pair of
Same side interior angles is _____________.
supplementary 1 2 3 4 5 7 6 8

10. Parallel Lines and Transversals
Theorem 3-3
Alternate
Exterior
Angles
If two parallel lines are cut by a transversal, then each pair of
alternate exterior angles is _________.
congruent 1 2 3 4 5 7 6 8 ?

11. End of Lesson

12. Transversals and Corresponding Angles
What You'll Learn
You will learn to identify the relationships among pairs of
corresponding angles formed by two parallel lines and a
transversal.

13. Transversals and Corresponding Angles
When a transversal crosses two lines, the intersection creates a number of angles that are related to each other.
Note 1 and 5 below. Although one is an exterior angle and the other is an interior angle, both lie on the same side of the transversal.
Angle 1 and 5 are called __________________.
corresponding angles l m 1 2 3 4 5 7 6 8 t
Give three other pairs of corresponding angles that are formed:
4 and 8
3 and 7
2 and 6

14. Transversals and Corresponding Angles
Postulate 3-1
Corresponding
Angles
If two parallel lines are cut by a transversal, then each pair of
corresponding angles is _________.
congruent

15. Transversals and Corresponding Angles
Concept
Summary
Congruent
Supplementary
Types of angle pairs formed when
a transversal cuts two parallel lines.
alternate interior
Same side interior
alternate exterior
corresponding

16. Transversals and Corresponding Angles1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
s || t and c || d.
Name all the angles that are congruent to 1.
Give a reason for each answer.
3  1
corresponding angles
6  1
vertical angles
8  1
alternate exterior angles
9  1
corresponding angles
14  1
alternate exterior angles
11  9  1
corresponding angles
16  14  1
corresponding angles

17. End of Lesson

18. Proving Lines Parallel
What You'll Learn
You will learn to identify conditions that produce parallel lines.
Reminder: In Chapter 1, we discussed “if-then” statements (pg. 24).
Within those statements, we identified the “__________” and the
“_________”.
hypothesis
conclusion
I said then that in mathematics, we only use the term
“if and only if”
if the converse of the statement is true.

19. Proving Lines Parallel
Postulate 3 – 1 (pg. 116):
IF ___________________________________,
THEN ________________________________________.
two parallel lines are cut by a transversal
two parallel lines are cut by a transversal
each pair of corresponding angles is congruent
each pair of corresponding angles is congruent
The postulates used are the converse of postulates that you already
know.
COOL, HUH?
, Postulate 3 – 2 (pg. 122):
IF ________________________________________,
THEN ____________________________________.

20. Proving Lines Parallel
Postulate 3-2
In a plane, if two lines are cut by a transversal so that a pair
of corresponding angles is congruent, then the lines are
_______.
parallel 1 2 a b
If 1 2,
then _____
a || b

21. Proving Lines Parallel
Theorem 3-3
In a plane, if two lines are cut by a transversal so that a pair
of alternate interior angles is congruent, then the two lines are _______.
parallel 1 2 a b
If 1 2,
then _____
a || b

22. Proving Lines Parallel
Theorem 3-6
In a plane, if two lines are cut by a transversal so that a pair
of alternate exterior angles is congruent, then the two lines are _______.
parallel 1 2 a b
If 1 2,
then _____
a || b

23. Proving Lines Parallel
Theorem 3-4
In a plane, if two lines are cut by a transversal so that a pair
of same side interior angles is supplementary, then the two lines are _______.
parallel 1 2 a b
If 1 + 2 = 180,
then _____
a || b

24. Proving Lines Parallel
Theorem 3-5
In a plane, if two lines are cut by a transversal so that a pair
of same side interior angles is supplementary, then the two lines are _______.
parallel
If a  t and b  t,
then _____ a b t
a || b

25. Proving Lines Parallel
We now have five ways to prove that two lines are parallel.
Concept
Summary
Show that a pair of corresponding angles is congruent.
Show that a pair of alternate interior angles is congruent.
Show that a pair of alternate exterior angles is congruent.
Show that a pair of same side interior angles is supplementary.
Show that two lines in a plane are perpendicular to a third line.

26. Proving Lines Parallel
Identify any parallel segments. Explain your reasoning. G A Y D R
90°

27. Proving Lines Parallel
Find the value for x so BE || TS. E B S T
(6x - 26)°
(2x + 10)°
(5x + 2)°
ES is a transversal for BE and TS.
mBES + mEST = 180
BES and EST are _________________ angles.
(2x + 10) + (5x + 2) = 180
(same side interior
7x = 180
If mBES + mEST = 180, then BE || TS by Theorem 4 – 7.
7x = 168
x = 24
Thus, if x = 24, then BE || TS.

28. End of Lesson