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# Lesson 3

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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