Download Geometry Notes T – 1: Parallel Lines and Transversals Definition

Geometry Notes T – 1: Parallel Lines and Transversals
Definition: Two coplanar lines are parallel if they never intersect
.
Postulate: Through a point not on a given line, there
is exactly one line parallel to the given line.
t
Vocabulary (Know these!)
1
2
Transversal: A line, t, that intersects two other lines, l1 and l2,
at different points.
Corresponding angles: Angles in the same relative positions.
Ex: 1 and 5 (both in the “upper right”)
Also, 2 and 6, 3 and 7, and 4 and 8.
6
7
l1
4
3
5
l2
8
Alternate interior angles: Angles between the two lines and on opposite
sides of the transversal.
Ex: 3 and 5, 4 and 6
Same side interior angles: Angles between the two lines and on the same side of the transversal.
Ex: 3 and 6, 4 and 5
Two Facts
1. If a line is translated in its own direction, its image will be the same line.
P
P'
2.
If two lines, l1 and l2, are parallel and P is any point on l1 and Q is any point on
l2, then after a translation along the vector PQ , the image of l1will coincide with
l2. (If l1 and l2, are not parallel , then no translation will make the image of l1
coincide with l2.)
P
Q
l1
l2
Corresponding Angles Theorem and Converse
Given: l1 and l2, transversal t intersects l1 at P and l2 at Q
a. If l1 || l2, then under the translation along PQ ,
the image of t is
the image of l1 is
The is means that the image of 1 is
Therefore, 1
, 2
, the image of 2 is
, etc.
, etc. because in each pair
b. If l1  l2, then under the translation along PQ , the image of t will still be t but the image of l1 will not be l2 and
so the image of 1 will not be 5 and the angles will not be congruent. (Same for the other three pairs.)
Theorems: When parallel lines are cut by a transversal,
and (converse)
When two lines are cut by a transversal and corresponding angles are congruent,
t
Ex: If l1 || l2, find the measures of all seven unknown angles on the diagram.
130 50
50 130
50
50 130
130
t
Ex: Which lines are parallel?
47
Lines b and d b/c alt. int. s are .
48
49
48
a
b
c
d
l1
l2
Alternate Interior Angles Theorem and Converse
Theorem: When parallel lines are cut by a transversal, alternate interior angles are congruent.
t
Given: l1 || l2, transversal t
1  5
Prove:
2
3
6
7
Statement
1
l1
4
5
l2
8
Reason
1. l1 || l2, transversal t
1. Given
2. 1  5
2. When lines are ||, corr. s are 
3. 1  3
3. Intersecting lines form vert. s which are 
4. 3  5
4. Transitive (2, 3)
The converse of this theorem is also true: When two lines are cut by a transversal and alternate interior angles
are congruent, the lines are parallel.
Ex: In the diagram at right, find the measures of the three marked angles and
determine if l1 || l2.
6x – 4 15x – 5
Note: We do not know if the lines are parallel. Do NOT
assume alternate interior angles are congruent.
5x + 5
We KNOW: Two adjacent angles that form a straight line
are supplementary. So
6x – 4 + 15x – 5 = 180
21x – 9 = 180
x=9
Then 6x – 4 = 6(9) – 4 = 50 and 5x + 5 = 5(9) + 5 = 50
Yes, the lines are parallel because alternate interior angles are congruent.
l2
l1