Parallel Lines & Transversals

1. Parallel Lines & Transversals

2. Definitions • Parallel lines – Two lines are parallel lines if they are coplanar and do not intersect. • Perpendicular lines – two lines are perpendicular if they intersect at a right angle. • Skew lines—Lines that do not intersect and are not coplanar.

3. Parallel Lines • Two lines are parallel lines if they are coplanar and do not intersect. m l

4. Perpendicular Lines • Two lines are perpendicular if they intersect at a right angle. • Perpendicular symbol: ┴ • Line m ┴ line l m l

5. Complementary Angles If the sum of the measures of two angles is exactly 90º then the angles are complementary.

6. Supplementary Angles If the sum of the measures of two angles is exactly 180º then the angles are supplementary.

7. Adjacent Angles • Two angles that share a common vertex and a common side, but no common interior points. (next to each other – share a side) • A & B are adjacent B A ●

8. Vertical Angles Vertical angles are formed by two intersecting lines. They are the angles that are opposite each other. Vertical angles are congruent (they have equal measures). They have the same vertex but do not share a side. In this picture, <1 and <3 are vertical angles. What is the measure of <2? <2 is 120º because it is congruent to the vertical angle across from it.

9. Transversals • A transversal crosses two or more lines at different points: • Interior angles are on either side of a transversal between a pair of lines. • Exterior angles are on either side of a transversal outside of a pair of lines. Angles 3,4,5, and 6 are interior angles. Angles 1,2,7, and 8 are exterior angles.

10. Interior Angles: • Alternate Interior Angles - Interior angles are on either side of a transversal between a pair of lines. • 35 and 46 • Same Side Interior Angles - Interior angles are on the same side of a transversal between a pair of lines. • 4 & 5 and 3 & 6 are not congruent, they are supplementary = 180°

11. Alternate Interior s • If 2  lines are cut by a transversal, then the pairs of alternate interior s are . • i.e. If l m, then 12. 1 2 l m

12. Same Side Interior s • If 2  lines are cut by a transversal, then the pairs of consecutive int. s are supplementary. • i.e. If l m, then 1 & 2 are supp. l m 1 2

13. Exterior Angles: • Alternate Exterior Angles-two angles that are formed by 2 lines and a transversal. These angles are outside the two lines on opposite sides of the transversal. • 17 and 28 • Same Side Exterior Angles - Two angles that are formed by 2 lines and a transversal. These angles lie outside the 2 lines on the same side of the transversal. • 1 & 8 and 2 & 7 are not congruent, they are supplementary = 180°

14. Alternate Exterior s • If 2  lines are cut by a transversal, then the pairs of alternate exterior s are . • i.e. If l m, then 12. l m 1 2

15. Corresponding Angles • Angles formed by a transversal cutting two or more lines and that are in the same relative position 1 ≅ 2 1 3 2 4 3 ≅ 4

16. Corresponding s • If 2  lines are cut by a transversal, then the pairs of corresponding s are . • i.e. If l m, then 34. 3 4 l m

17. 1 2 4 3 5 6 8 7 Examples: • Transversal: a line that intersects two or more coplanar lines at different points. • Angles 1 and 5 are corresponding angles • Angles 1 and 7 are alternate exterior angles • Angles 3 and 5 are alternate interior angles. • 3 and 6 are same side interior angles

18. Identifying Angle relationships • List all pairs of angles that fit the description. • Corresponding • Alternate Exterior • Alternate Interior • Same Side Interior • Same side Exterior 2 4 1 6 8 3 5 7

19. Solution: • 1 and 5; 2 and 6; 3 and 7, 4 and 8 • 1 and 8, 2 and 7 • 3 and 6, 4 and 5 • 3 and 5, 4 and 6 • 1 and 7, 2 and 8

20. Ex: Find: m1= m2= m3= m4= m5= m6= x= 1 125o 2 3 5 4 6 x+15o

21. Ex: Find: m1=55° m2=125° m3=55° m4=125° m5=55° m6=125° x=40° 1 125o 2 3 5 4 6 x+15o