Angle pairs formed by parallel lines cut by a transversal

When two parallel lines are given in a figure, there are two main areas: the interior and the exterior.

When two parallel lines are cut by a third line, the third line is called the transversal. In the example below, eight angles are formed when parallel lines m and n are cut by a transversal line, t.

There are several special pairs of angles formed from this figure. Some pairs have already been reviewed:
      Vertical pairs:       angle1 and angle4
                                  angle2 and angle3
                                  angle5 and angle8
                                  angle6 and angle7

Recall that all pairs of vertical angles are congruent.
      Supplementary pairs:       angle1 and angle2
                                            angle2 and angle4
                                            angle3 and angle4
                                            angle1 and angle3
                                            angle5 and angle6
                                            angle6 and angle8
                                            angle7 and angle8
                                            angle5 and angle7
Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs. There are other supplementary pairs described in the shortcut later in this section. There are three other special pairs of angles. These pairs are congruent pairs.

Alternate interior angles two angles in the interior of the parallel lines, and on opposite (alternate) sides of the transversal. Alternate interior angles are non-adjacent and congruent.


Alternate exterior angles two angles in the exterior of the parallel lines, and on opposite (alternate) sides of the transversal. Alternate exterior angles are non-adjacent and congruent.


Corresponding angles two angles, one in the interior and one in the exterior, that are on the same side of the transversal. Corresponding angles are non-adjacent and congruent.

Use the following diagram of parallel lines cut by a transversal to answer the example problems.

Example:
What is the measure of angle8?
The angle marked with measure 53° and angle8 are alternate exterior angles. They are in the exterior, on opposite sides of the transversal. Because they are congruent, the measure of angle8 = 53°.
Example:
What is the measure of angle7?
angle8 and angle7 are a linear pair; they are supplementary. Their measures add up to 180°. Therefore, angle7 = 180° – 53° = 127°.

1. When a transversal cuts parallel lines, all of the acute angles formed are congruent, and all of the obtuse angles formed are congruent.

In the figure above angle1, angle4, angle5, and angle7 are all acute angles. They are all congruent to each other. angle1 ≅ angle4 are vertical angles. angle4 ≅ angle5 are alternate interior angles, and angle5 ≅ angle7 are vertical angles. The same reasoning applies to the obtuse angles in the figure: angle2, angle3, angle6, and angle8 are all congruent to each other.

2. When parallel lines are cut by a transversal line, any one acute angle formed and any one obtuse angle formed are supplementary.

From the figure, you can see that angle3 and angle4 are supplementary because they are a linear pair.
Notice also that angle3 ≅ angle7, since they are corresponding angles. Therefore, you can substitute angle7 for angle3 and know that angle7 and angle4 are supplementary.

Example:
In the following figure, there are two parallel lines cut by a transversal. Which marked angle is supplementary to angle1?

The angle supplementary to angle1 is angle6. angle1 is an obtuse angle, and any one acute angle, paired with any obtuse angle are supplementary angles. This is the only angle marked that is acute.




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