2. LINES AND ANGLES Definitions Free powerpoints at http://www.worldofteaching.comhttp://www.worldofteaching.com Modified by Lisa Palen

3. PARALLEL LINES Definition: Parallel lines are coplanar lines that do not intersect. Illustration: Use arrows to indicate lines are parallel. Notation: || means “is parallel to.” l || m AB || CD l m A B C D

4. PERPENDICULAR LINES Definition: Perpendicular lines are lines that form right angles. Illustration: Notation: m  n Key Fact: 4 right angles are formed. m n

5. OBLIQUE LINES Definition: Oblique lines are lines that intersect, but are NOT perpendicular. Illustration: Notation: m and n are oblique.

6. SKEW LINES Two lines are skew if they do not intersect and are not in the same plane (They are noncoplanar).

7. PARALLEL PLANES All planes are either parallel or intersecting. Parallel planes are two planes that do not intersect.

8. EXAMPLES 1.Name all segments that are parallel to 2.Name all segments that intersect 3.Name all segments that are skew to 4.Name all planes that are parallel to plane ABC. Answers: 1.Segments EH, BC & GF. 2.Segments AE, AB, DH & DC. 3.Segments CG, BF, FE & GH. 4.Plane FGH.

9. Recall: Slope measures how steep a line is. The slope of the non-vertical line through the points (x 1, y 1 ) and (x 2, y 2 ) is slope Review of Slope

10. If a line goes up from left to right, then the slope has to be positive. Conversely, if a line goes down from left to right, then the slope has to be negative.

11. Examples Find the slope of the line through the given points and describe the line. (rises to the right, falls to the right, horizontal or vertical.) 1)(1, -4) and (2, 5)2)(5, -2) and (- 3, 1) Solution slope Solution slope This line rises to the right. This line falls to the right.

12. The slope of a horizontal line is zero. The slope of a vertical line is undefined. Sometimes we say a vertical line has no slope.

13. 3)(7, 6) and (-4, 6)4)(-3, -2) and (-3, 8) Solution slope More Examples Find the slope of the line through the given points and describe the line. (rises to the right, falls to the right, horizontal or vertical.) This line is horizontal. This line is vertical. Solution slope No division by zero!

14. Horizontal lines have a slope of zero while vertical lines have undefined slope. HorizontalVertical m = 0 m = undefined

15. Slopes of Parallel lines Postulate (Parallel lines have equal slopes.) Two non-vertical lines are parallel if and only if they have equal slopes. Also: All horizontal lines are parallel. All vertical lines are parallel. All lines with undefined slope are parallel. (They are all vertical.)

16. x y Slopes of Parallel lines Like this? Or this? ____ Example What is the slope of this line? What is the slope of any line parallel to this line? 5/12 5/12 because parallel lines have the same slope!

17. Slopes of Perpendicular lines Postulate Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. The slopes of non-vertical perpendicular lines are negative reciprocals. m and or and

18. Undefined ! Slopes of Perpendicular lines Examples Find the negative reciprocal of each number: 1. 2. 3. 4.

19. Slopes of Perpendicular Lines 0 and Also All horizontal lines are perpendicular to all vertical lines. The slope of a line perpendicular to a line with slope 0 is undefined. Undefined !

20. Examples Any line parallel to a line with slope has slope _____. Any line perpendicular to a line with slope has slope _____. Any line parallel to a line with slope 0 has slope _____. Any line perpendicular to a line with undefined slope has slope _____. Any line parallel to a line with slope 2 has slope _____.

21. Def: a line that intersects two lines (that are coplanar) at different points Illustration: Transversal t 30

22. Vertical Angles Two non-adjacent angles formed by intersecting lines. They are opposite angles. 12 3 4 5 6 78 t 1   41   4  2   3  5   8  6   7

23. Vertical Angles Find the measures of the missing angles. 125  xx yy 55  t 125  x = 125 y = 55

24. Linear Pair Supplementary adjacent angles. They form a line and their sum = 180  ) 12 3 4 5 6 78 t m  1 + m  2 = 180º m  2 + m  4 = 180º m  4 + m  3 = 180º m  3 + m  1 = 180º m  5 + m  6 = 180º m  6 + m  8 = 180º m  8 + m  7 = 180º m  7 + m  5 = 180º

25. Supplementary Angles/ Linear Pair Find the measures of the missing angles. xx 72  yy t 108  x = 180 – 72 y = x = 108

26. Corresponding Angles Two angles that occupy corresponding positions. Top Left t Top Right Bottom Right Bottom Left 11 and  5 22 and  6 33 and  7 44 and  8 1 2 34 5 6 78

27. Corresponding Angles Postulate If two parallel lines are crossed by a transversal, then corresponding angles are congruent. Top Left t Top Right Bottom Right Bottom Left 11   5 22   6 33   7 44   8 1 2 34 5 6 78

28. Corresponding Angles Find the measures of the missing angles, assuming the black lines are parallel. 145  zz t z = 145

29. Alternate Interior Angles Two angles that lie between the two lines on opposite sides of the transversal t 33 and  6 44 and  5 1 2 34 5 6 78

30. If two parallel lines are crossed by a transversal, then alternate interior angles are congruent. Alternate Interior Angles Theorem t 33   6 44   5 1 2 34 5 6 78

31. If two parallel lines are crossed by a transversal, then alternate interior angles are congruent. Alternate Interior Angles Theorem t 1 4 5 l m Given: l  m Prove:  4   5 Statements Reasons 1.l  m 2.  4   1 3.  1   5 4.  4   5 1.Given 2.Vertical AnglesThm 3. Corres- ponding Angles Post. 4. Transitive Property of Congruence

32. Alternate Interior Angles Find the measures of the missing angles, assuming the black lines are parallel. 82  z z  t z = 82

33. Alternate Exterior Angles Two angles that lie outside the two lines on opposite sides of the transversal t 22 and  7 11 and  8 1 2 34 5 6 78

34. Alternate Exterior Angles Theorem If two parallel lines are crossed by a transversal, then alternate exterior angles are congruent. t 22   7 11   8 1 2 34 5 6 78

35. Alternate Exterior Angles Find the measures of the missing angles, assuming the black lines are parallel. 120  ww t w = 120

36. Consecutive Interior Angles Two angles that lie between the two lines on the same sides of the transversal t m  3 and m  5 m  4 and m  6 1 2 34 5 6 78

37. If two parallel lines are crossed by a transversal, then consecutive interior angles are supplementary. Consecutive Interior Angles Theorem t m  3 +m  5 = 180º m  4 +m  6 = 180º 1 2 34 5 6 78

38. Consecutive Interior Angles Find the measures of the missing angles, assuming the black lines are parallel. ? t 135  45  180º - 135º

39. Angles and Parallel Lines If two parallel lines are crossed by a transversal, then the following pairs of angles are congruent. Corresponding angles Alternate interior angles Alternate exterior angles If two parallel lines are crossed by a transversal, then the following pairs of angles are supplementary. Consecutive interior angles

40. Review Angles and Parallel Lines Alternate interior angles 1 2 34 56 78 Alternate exterior angles Corresponding angles Consecutive interior angles Consecutive exterior angles

41. Examples If line AB is parallel to line CD and s is parallel to t, find the measure of all the angles when m  1 = 100º. Justify your answers. m  2=80º m  3=100º m  4=80º m  5=100º m  6=80º m  7=100º m  8=80º m  9=100º m  10=80º m  11=100º m  12=80º m  13=100º m  14=80º m  15=100º m  16=80º Click for Answers

42. More Examples 1. The value of x, if m  3 = (4x + 6)º and the m  11 = 126º. If line AB is parallel to line CD and s is parallel to t, find: 2. The value of x, if m  1 = 100º and m  8 = (2x + 10)º. 3. The value of y, if m  11 = (3y – 5)º and m  16 = (2y + 20)º. ANSWERS: 1. 302. 353. 33 Click for Answers

43. Proving Lines Parallel Recall: Corresponding Angles Postulate If two lines cut by a transversal are parallel, then corresponding angles are congruent. So what can you say about the lines here? Contrapositive: If corresponding angles are NOT congruent, then two lines cut by a transversal are NOT parallel. 145  144  NOT PARALLEL!

44. So what can you say about the lines here? Corresponding Angles Postulate: If two lines cut by a transversal are parallel, then corresponding angles are congruent. Converse of the Corresponding Angles Postulate: If corresponding angles are congruent, then two lines cut by a transversal are parallel. Proving Lines Parallel 144  PARALLEL!

45. If corresponding angles are congruent, then two lines cut by a transversal are parallel. Proving Lines Parallel Converse of the Corresponding Angles Postulate two lines cut by a transversal are parallel

46. Proving Lines Parallel Converse of the Alternate Interior Angles Theorem If alternate interior angles are congruent, then two lines cut by a transversal are parallel.

47. If alternate interior angles are congruent, then two lines cut by a transversal are parallel. Converse of the Alternate Interior Angles Theorem t 1 4 5 l m Given:  5   4 Prove: l  m Statements Reasons 1.  5   4 2.  4   1 3.  1   5 4.l  m 1.Given 2.Vertical AnglesThm 3. Transitive Property of Congruence 4. Converse of the Corr- esponding Angles Post/

48. Proving Lines Parallel Converse of the Alternate Exterior Angles Theorem If alternate exterior angles are congruent, then two lines cut by a transversal are parallel.

49. Proving Lines Parallel Converse of the Consecutive Interior Angles Theorem If consecutive interior angles are SUPPLEMENTARY, then two lines cut by a transversal are parallel.

50. Proving Lines Parallel Examples Find the value of x which will make lines a and lines b parallel. 1.2. 3. 4. ANSWERS: 20; 50; 45; 20

51. Ways to Prove Two Lines Parallel Show that corresponding angles are congruent. Show that alternative interior angles are congruent. Show that alternate exterior angles are congruent. Show that consecutive interior angles are supplementary. In a plane, show that the lines are perpendicular to the same line.