parallel lines initial definitions and theorems
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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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Interior angles are angles that are between the lines.
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Interior angles are angles that are between the lines.
3, 4, 5, and 6 are interior angles.
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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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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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Corresponding angles are angles that are in the same relative positions when compared to the points of intersection.
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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.
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m n
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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.
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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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m n
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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.
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m n
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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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Alternate interior angles are interior angles that are on opposite sides of the transversal.
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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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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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Alternate exterior angles are exterior angles that are on opposite sides of the transversal.
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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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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.
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Parallel lines are lines that will never intersect, no matter how far we extend them.
We can write m||n.
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When m and n are parallel, we get many nice relationships between the pairs of angles.
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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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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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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.
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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.
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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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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.