Do Now:

1
Do Now
B
A
Given -XB bisects AXC -angle EXC is a
straight angle. Find (not prove) -Find
measure of angle BXC and angle BXA
C
75
X
D
E
2
Geometry
THE FATE OF THE UNIVERSE!
A.K.A. How can we show that two lines are
parallel?
3

• What are Parallel Lines?

Definition Parallel lines are coplanar (in the
same plane) and either have NO points in common
or every point in common (like the lines are on
top of one another).
4
Postulates relating to Parallel Lines

• Postulate A line is parallel to itself
• (reflexive property)

Postulate If two lines are each parallel to the
same line, they are parallel to each
other. (transitive property)
5

• What is a transversal?

Definition A transversal is a line that
intersects two other lines in two DIFFERENT
POINTS.
This is NOT a transversal.
This IS a transversal.
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What can we say about the angles formed by a
transversal?
What are the exterior angles?
1, 2, 7, and 8
What are the interior angles?
3, 4, 5, and 6
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Alternate Interior Angles
Interior angles on OPPOSITE sides of the
transversal at different vertices.
The alternate interior angles are
4 and 6
3 and 5
THESE COME IN PAIRS!
4 and 5 are NOT alternate interior angles.
8
Corresponding Angles

• Angles in the same position, but at different
  vertices along the transversal.

Examples
1 and 5
2 and 6
3 and 7
5 and 6 are NOT corresponding angles
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Do Now
Identify at least one pair of angles that are

1. Corresponding angles
2. Alternate Interior Angles
3. Supplementary angles
4. Vertical angles

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Aim What is important about Corresponding and
Alternate Interior angles?

• Theorem If a transversal cuts (crosses) two
  parallel lines, the alternate interior angles are
  congruent.
• Theorem If a transversal cuts two parallel
  lines, then corresponding angles are congruent.

11
2
1
4
3
5
6
7
8
What angles are congruent to angle 5 here?
7 (Vertical Angle)
3 (Alternate Interior Angle)
1 (Corresponding Angle)
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The Converse of both these theorems are true, too!

• Theorem If a transversal crosses two lines, and
  the alternate interior angles are congruent, then
  the line are parallel.

Theorem If a transversal crosses two lines, and
the corresponding angles are congruent, then the
line are parallel.
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Theorem
IF A TRANSVERSAL CROSSES TWO PARALLEL LINES

• The interior angles on the same side of the
  transversal are supplementary.
• The exterior angles on the same side of a
  transversal are supplementary.

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These theorems tell us that
These angle pairs are supplementary
4 5
3 6
1 8
2 7