Transcript Geometry

Do Now:
B
A
Given:
C
-XB bisects AXC
-angle EXC is a
straight angle.
X
E
75
°
Find (not prove):
D
-Find measure of angle
BXC and angle BXA
Geometry
&
A.K.A.: How can we show that two lines are parallel?
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).
A
B
C
D
Notation:
AB || CD
Line AB is parallel to line CD
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)
What is a transversal?
Definition: A transversal is a line that intersects two
other lines in two DIFFERENT POINTS.
This IS a transversal.
This is NOT a transversal.
What can we say about the
angles formed by a
transversal?
Exterior
Interior
What are the exterior angles?
1, 2, 7, and 8
What are the interior angles?
3, 4, 5, and 6
2 1
3 4
6 5
7 8
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.
2 1
3 4
6 5
7 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
2 1
3 4
6 5
7 8
Do Now
Identify at least one pair of angles that are
a) Corresponding angles
b) Alternate Interior Angles
c) Supplementary angles
d) Vertical angles
2 1
3 4
6 5
7 8
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.
2 1
3 4
6
7
5
8
This is how we show
on a diagram
that two lines are parallel
What angles are congruent to angle 5 here?
7 (Vertical Angle)
3 (Alternate Interior Angle)
1 (Corresponding Angle)
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.
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.
These theorems tell us that…
These angle pairs are supplementary:
4&5
3&6
1&8
2&7
2 1
3 4
6 5
7 8