Download Parallel Lines cut by a Transversal

Parallel Lines cut by a
Transversal
Geometry Standard 7.0
Prove & use theorems involving the
properties of parallel lines cut by a
transversal.
Parallel Lines Cut by a
Transversal
m
j
k
Lines j & k are parallel. They are cut by the transversal
line m.
Parallel lines ?
Parallel Lines?
Parallel Lines Cut by
Parallel Lines
Parallel Fields
Parallel Walls
Parallel Church
Parallel City
Parallel π’s?
Parallel Lines in a Quilt
Parallel Lines Cut by a
Transversal
Behavior of coplanar lines and slopes
• Intersecting lines have different slopes.
4
2
-5
5
-2
-4
-6
• Parallel lines have the same slope.
4
2
-5
5
-2
-4
-6
Huh?
What this comes down to is the following:
If angles 1 and 2 have measures that add up to
less than 180 degrees, then the non-transversal
lines forming them will intersect.
1
2
So, the Parallel Postulate states the converse
How many Parallel lines do you see?
How many transversals?
Are stair steps always parallel?
Parallel, Skew or…?
Which are the Parallel Lines?
Which are the Transversals?
Things are getting a little “stair” crazy!
And more so!!!
So, in the terms of high school
geometry:
Given a line & a point not on the line, there is exactly
one line in the plane of the given line, that goes through
the point parallel to the line.
So, on a plane, when parallel lines are
crossed by a transversal, what happens?
m
1
j
2
4
3
5
k
6
8
7
Names of Associated Angles
m
j
2
1
3
4
5 6 7
8
k
• Alternate Exterior Angles are on
opposite sides of the transversal
– 1 &  7
– 2 &  8
•Corresponding Angles lie on the
same side of the transversal situated
the same way on two different
parallel lines. (Angles 2 & 6, 3 & 7, 1
& 5, 4 & 8
• Interior Angles lie between
the two lines
– 3, 4 ,5, 6
• Alternate Interior Angles are
on opposite sides of the
transversal
–
–
 3 &5
 4 &6
• Consecutive Interior Angles
are on the same side of the
transversal
–  3 &6
–  4 &5
• Exterior Angles lie outside the
two lines
–
 1,  2,  7,  8
What do we know?
m
1
j
k
2
3
4
5 6 7
8
• Vertical angles are
congruent.
– 1 &  3
– 2 &  4
– 5 &  7
– 6 &  8
• Linear pairs form
supplementary angles.
–  1 & 2
–  2 & 3
–  3 & 4
–  4 & 1
–  5 & 6
–  6 & 7
–  7 & 8
–  8 & 1
When two parallel lines are cut by a
transversal, the following statements are true:
•
•
•
•
Corresponding angles are congruent.
Alternate interior angles are congruent.
Alternate exterior angles are congruent.
Interior angles on the same side of the transversal are
supplementary.
• If a transversal is perpendicular to one of the two
parallel lines, then it is automatically parallel to the
other line
m
j
k
1
2
3
4
6
5
7
8
Theorems or Postulates?
• Choose one of the properties on the last slide to
be a postulate & then you can prove all the
others by applying it.
– Postulate: If two parallel lines are cut by a
transversal, then corresponding angles are
congruent.
m
1
j
k
2
3
4
5 6 7
8
1  R5
R2  R6
R4  R8
R3  R7
Alternate Interior Angle Theorem: It two parallel
lines are cut by a transversal, then alternate
interior angles are congruent.
m
• Given: j || m
• Prove: 3   5 &  4   6
2
1
j
k
4
3
5 6 7
8
Statements
1. j || m
2. 1  R5, R2  R6,
R4  R8, R3  R7
3. 1  R3, R5  R7
R6  R8, R2  R4
4. 3  R5, R4  R6
Reasons
1.
2.
3.
4.
Alternate exterior Theorem: If two parallel lines
are cut by a transversal then the alternating
exterior angles are congruent.
m
1
3
q
2
4
Given: m || n, q is a transversal;
Prove:
2  R7, R1  R8
5 6
7 8
We have proven that when parallel lines are cut by a transversal, alternate interior angles are congruent. So we know
that angles 3 & 6 & angles 4 & 5 are congruent. Since angles 2
& 3, angles 1 & 4, angles 5 & 8 & angle 6 & 7 are vertical
angles, they are also congruent. So by the transitive property
applied twice, angles 2 & 7 & angles 1 & 8 are congruent.
n
Consecutive Angles Theorem: If two parallel lines
are cut by a transversal then the consecutive
interior angles formed are supplementary