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
.
Postulate: Through a point not on a given line, there
is exactly one line parallel to the given line.
t
Vocabulary (Know these!)
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.
1
2
Transversal: A line, t, that intersects two other lines, l1 and l2,
at different points.
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
t
Given: l1 and l2, transversal t intersects l1 at P and l2 at Q
2
3
a. If l1 || l2, then under the translation along PQ ,
6
7
the image of t is
1
l1
P
4
5
Q
8
l2
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.
l1
130
t
Ex: Which lines are parallel?
47
48
49
48
Alternate Interior Angles Theorem and Converse
a
b
c
d
l2
Theorem: When parallel lines are cut by a transversal, alternate interior angles are congruent.
t
Given:
2
Prove:
3
6
7
1
l1
4
5
l2
8
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
5x + 5
l2
l1
Geometry HW: Triangles - 1
1. The Alternate Interior Angle Theorem can also be proved using a rigid motion.
In the diagram at right, name a single transformation after which the image of
3 will be 5 and the image of 4 will be 6. Be specific. (We’re not going
to do the actual proof because some of the details get very confusing.)
t
P
l1
3 4
6 5
l2
Q
2. We wish to prove the following theorem: If parallel lines are cut by a transversal, same side interior angles
are supplementary.
t
Given: l1 || l2, transversal t
1
Prove: 3 and 2 are supplementary
l1
2
On your paper, fill in the missing statements and reasons labeled (a) – (g).
3
Statement
1. l1 || l2, transversal t
2. 1 and 2 are supplementary
3. 1 + 2 = 180
4. 1  3
5. (d)
6. (f)
Reason
1. Given
2. (a)
3. (b)
4. (c)
5. (e)
6. (g)
l2
a
115
125
3. In the diagram at right, which lines are parallel?
(1) a and b, only
(2) a and c, only
(3) b and c only
(4) All three of them
(5) None of them
65
b
)
c
4. Each diagram shows two parallel lines cut by a transversal. Find the value of x in each diagram.
a.
b.
(x2
c.
 30)
(3x – 12)
(5x + 6)
5. In the diagram at right, l1 || l2 and each algebraic expression
represents an angle.
a. Determine the value of x.
l1
l2
l3
(4x + 10) (8x + 5)
(5x – 4)
l4
b. Determine if l3 || l4 and give a reason.
6. In the diagram at right, if l1 || l2, find the values of x and y.
l1
(2x)
(x + 3y) (3x – 2y)
l2