Lesson 3 Parallel Lines

Definition Two lines are parallel if they lie in the same plane and do not intersect. If lines m and n are parallel we write p q p and q are not parallel m n

We also say that two line segments, two rays, a line segment and a ray, etc. are parallel if they are parts of parallel lines. D C A B P m Q

The Parallel Postulate Given a line m and a point P not on m, there is one and no more than one line that passes through P and is parallel to m. P m

Transversals A transversal for lines m and n is a line t that intersects lines m and n at distinct points. We say that t cuts m and n. A transversal may also be a line segment and it may cut other line segments. E m A B n C D t

We will be most concerned with transversals that cut parallel lines. When a transversal cuts parallel lines, special pairs of angles are formed that are sometimes congruent and sometimes supplementary.

Corresponding Angles A transversal creates two groups of four angles in each group. Corresponding angles are two angles, one in each group, in the same relative position. 1 2 m 3 4 5 6 n 7 8

Alternate Interior Angles When a transversal cuts two lines, alternate interior angles are angles within the two lines on alternate sides of the transversal. m 1 3 4 2 n

Alternate Exterior Angles When a transversal cuts two lines, alternate exterior angles are angles outside of the two lines on alternate sides of the transversal. 1 3 m n 4 2

Interior Angles on the Same Side of the Transversal When a transversal cuts two lines, interior angles on the same side of the transversal are angles within the two lines on the same side of the transversal. m 1 3 2 4 n

Exterior Angles on the Same Side of the Transversal When a transversal cuts two lines, exterior angles on the same side of the transversal are angles outside of the two lines on the same side of the transversal. 1 3 m n 2 4

Example In the figure and Find Since angles 1 and 2 are vertical, they are congruent. So, Since angles 1 and 3 are corresponding angles, they are congruent. So, 1 3 2 m n

Example In the figure, and Find Consider as a transversal for the parallel line segments. Then angles B and D are alternate interior angles and so they are congruent. So, A B C D E

Example In the figure, and If then find Considering as a transversal, we see that angles A and B are interior angles on the same side of the transversal and so they are supplementary. So, Considering as a transversal, we see that angles B and D are interior angles on the same side of the transversal and so they are supplementary. A B ? C D

Example In the figure, bisects and Find Note that is twice So, Considering as a transversal for the parallel line segments, we see that are corresponding angles and so they are congruent. A D E B C

Example In the figure is more than and is less than twice Also, Find m Let denote Then Note that angles 2 and 4 are alternate interior angles and so they are congruent. So, Adding 44 and subtracting from both sides gives So, Note that angles 1 and 5 are alternate interior angles, and so m n 2 4 3 5 1

Proving Lines Parallel So, far we have discussed that if we have a pair of parallel lines, then certain pairs of angles created by a transversal are congruent or supplementary. Now we consider the converse. If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. If the alternate interior or exterior angles are congruent, then the lines are parallel. If the interior or exterior angles on the same side of the transversal are supplementary, then the lines are parallel.

Example In the figure, angles A and B are right angles and What is Since these angles are supplementary. Note that they are interior angles on the same side of the transversal This means that Now, since angles C and D are interior angles on the same side of the transversal they are supplementary. So, A D B C

This is a nice fact to remember. Given a line m, In the previous example, there were two lines each perpendicular to a third, and we concluded that the two lines are parallel. This is a nice fact to remember. Given a line m, if p is perpendicular to m, and q is perpendicular to m, then p q m

Three Parallel Lines In the diagram, if and then m n p

Example In the figure, , and Find According to the parallel postulate, there is a line through E parallel to Draw this line and notice that this line is also parallel to Note that and are alternate interior angles and so they are congruent. So, Similarly, and so Therefore, C D 2 1 E A B