Module 3 - Parallel Lines

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m n
t
A transversal is a line that intersects two (or more) other lines at
distinct points.
Lines m and n are cut by transversal t.

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m n
t
Interior angles are angles that are between the lines.

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m n
t
Interior angles are angles that are between the lines.
∠3, ∠4, ∠5, and ∠6 are interior angles.

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m n
t
Interior angles are angles that are between the lines.
∠3, ∠4, ∠5, and ∠6 are interior angles.
Exterior angles are angles that are between the lines.

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m n
t
Interior angles are angles that are between the lines.
∠3, ∠4, ∠5, and ∠6 are interior angles.
Exterior angles are angles that are between the lines.
∠1, ∠2, ∠7, and ∠8 are exterior angles.

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m n
t
Corresponding angles are angles that are in the same relative
positions when compared to the points of intersection.

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m n
t
Corresponding angles are angles that are in the same relative
positions when compared to the points of intersection.
∠1 and ∠5 are a pair of corresponding angles because they are
both up and to the left of the points of intersection.

8. 1
2
3
4
5
6
7
8
m n
t
Corresponding angles are angles that are in the same relative
positions when compared to the points of intersection.
∠1 and ∠5 are a pair of corresponding angles because they are
both up and to the left of the points of intersection.

9. 1
2
3
4
5
6
7
8
m n
t
Corresponding angles are angles that are in the same relative
positions when compared to the points of intersection.
∠1 and ∠5 are a pair of corresponding angles because they are
both up and to the left of the points of intersection.
∠3 and ∠7 are a pair of corresponding angles because they are
both up and to the right of the points of intersection.

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2
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5
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m n
t
Corresponding angles are angles that are in the same relative
positions when compared to the points of intersection.
∠1 and ∠5 are a pair of corresponding angles because they are
both up and to the left of the points of intersection.
∠3 and ∠7 are a pair of corresponding angles because they are
both up and to the right of the points of intersection.
∠2 and ∠6 are a pair of corresponding angles because they are
both down and to the left of the points of intersection.

11. 1
2
3
4
5
6
7
8
m n
t
Corresponding angles are angles that are in the same relative
positions when compared to the points of intersection.
∠1 and ∠5 are a pair of corresponding angles because they are
both up and to the left of the points of intersection.
∠3 and ∠7 are a pair of corresponding angles because they are
both up and to the right of the points of intersection.
∠2 and ∠6 are a pair of corresponding angles because they are
both down and to the left of the points of intersection.
∠4 and ∠8 are a pair of corresponding angles because they are
both down and to the right of the points of intersection.

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m n
t
Alternate interior angles are interior angles that are on opposite
sides of the transversal.

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m n
t
Alternate interior angles are interior angles that are on opposite
sides of the transversal.
∠3 and ∠6 are alternate interior angles.

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m n
t
Alternate interior angles are interior angles that are on opposite
sides of the transversal.
∠3 and ∠6 are alternate interior angles.
∠4 and ∠5 are another pair of alternate interior angles.

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m n
t
Alternate exterior angles are exterior angles that are on
opposite sides of the transversal.

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m n
t
Alternate exterior angles are exterior angles that are on
opposite sides of the transversal.
∠1 and ∠8 are alternate exterior angles.

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m n
t
Alternate exterior angles are exterior angles that are on
opposite sides of the transversal.
∠1 and ∠8 are alternate exterior angles.
∠2 and ∠7 are another pair of alternate exterior angles.

18. m
n
Parallel lines are lines that will never intersect, no matter how
far we extend them.
We can write m||n.

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m
n
t
When m and n are parallel, we get many nice relationships
between the pairs of angles.

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m
n
t
When m and n are parallel, we get many nice relationships
between the pairs of angles.
Postulate: If two parallel lines are cut by a transversal, then
the corresponding angles are congruent.
So if m||n, then ∠1 and ∠5 are congruent.

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m
n
t
When m and n are parallel, we get many nice relationships
between the pairs of angles.
Theorem: If two parallel lines are cut by a transversal, then
the alternate interior angles are congruent.
So if m||n, then ∠3 and ∠6 are congruent.

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m
n
t
When m and n are parallel, we get many nice relationships
between the pairs of angles.
Theorem: If two parallel lines are cut by a transversal, then
the alternate exterior angles are congruent.
So if m||n, then ∠1 and ∠8 are congruent.

23. 1
2
3
4
5
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8
m
n
t
When m and n are parallel, we get many nice relationships
between the pairs of angles.
Theorem: If two parallel lines are cut by a transversal, then
the interior angles on the same side of the transversal are
supplementary.
So if m||n, then ∠3 and ∠5 are supplementary.

24. 1
2
3
4
5
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m
n
t
When m and n are parallel, we get many nice relationships
between the pairs of angles.
Theorem: If two parallel lines are cut by a transversal, then
the exterior angles on the same side of the transversal are
supplementary.
So if m||n, then ∠2 and ∠8 are supplementary.

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m n
t
It is very important to remember that these relationships are
only true when the lines are parallel.
In the figure, we see that m and n are not parallel and ∠1 and
∠5 are obviously not congruent.