1 / 22

Lesson 3

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.

adamdaniel
Download Presentation

Lesson 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lesson 3 Parallel Lines

  2. 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

  3. 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

  4. 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

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

  6. 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.

  7. 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. 2 1 m 3 4 6 5 n 8 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

  9. 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

  10. Interior Angles on the Same Sideof 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

  11. Exterior Angles on the Same Sideof 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 4 2

  12. 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 n m

  13. 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 E D

  14. 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. • So, A B ? C D

  15. 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. • So, A D E C B

  16. Example • In the figure is more than and is less than twice • Also, Find • 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

  17. 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.

  18. 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

  19. 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

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

  21. 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, D C 2 1 E A B

More Related