Chapter 3.1

1. Chapter 3.1 Parallel Lines and Transversals

2. Relationships Between Lines and Planes • Parallel Lines – The old definition was “lines that didn’t intersect” • This definition is now not good enough. Are lines l and m parallel? l What more do we need in the def? m

3. Parallel Lines • Parallel Lines are coplanar lines that do not intersect. • The symbol of parallel is this….||. • On diagrams we can use something like this to indicate that the lines are parallel.

4. Parallel Planes • Planes can be parallel. • Planes don’t have to be coplanar to be parallel. As a matter of fact, if planes are coplanar then they are the same plane. • Example of parallel planes. C B Plane AEHD is parallel to PlaneBFGC. Any others? D A G F E H

5. Skew Lines • Skew lines are lines that are not coplanar. • These lines can never be parallel. • Because they have to be coplanar to be parallel. • They can never intersect. • Because they have to be coplanar to intersect too. • To find Skew lines you need to find noncoplanar lines. • To do this you could eliminate all… • Parallel Lines • Coplanar Lines

6. B C D A G F E H Skew Lines Find all lines skew to line AE. 1) Find all lines parallel to line AE. 2) Eliminate all coplanar lines to line AE. All remaining lines are Skew to line AE.

7. B C D A G F E H Another Example Find all lines skew to line BC. 1) Find lines parallel to line BC 2) Find lines coplanar to line BC All remaining lines are skew to line BC! Lines DH, AE, EF and GH are skew to line BC!

8. Transversals • Transversal – A transversal is line that intersects two or more coplanar lines in different places. • Notice the word parallel is not in the definition. • When a transversal intersects two coplanar lines in two different places it creates EIGHT different angles.

9. Transversals (Con’t) Two coplanar lines. l 2 1 4 A transversal. 3 5 6 Eight angles. m 7 8 n Think of linesl and mas the rails of a railroad track and linenas the railroad tie.If you do, then you will see that the “choo-choo train will travel right or left on the rails.

10. l 2 1 4 3 5 6 m 7 8 n Transversals (Con’t) Thinking of the choo-choo train, which angles are interior of the “rails”? <3, <4, <5 & <6 Which angles are exterior of the “rails”? <1, <2, <7 & <8 Important angle pairs of angles made by a transversal intersecting a pair of lines. They are Alternate Interior, Alternate Exterior, ConsecutiveInterior, Consecutive Exterior and Corresponding.

11. Transversal and Angle Pairs • Alternate Interior Angle Pairs – • A pair of interior angles, one per vertex, that are on alternate sides of the transversal. • Alternate Exterior Angle Pairs – • A pair of exterior angles, one per vertex, that are on alternate sides of the transversal. • Consecutive Interior Angle Pairs – • A pair of interior angles, one per vertex, that are on the same side of the transversal. • Consecutive Exterior Angle Pairs – • A pair of exterior angles, one per vertex, that are on the same side of the transversal.

12. Transversal and Angle Pairs • Corresponding Angle Pairs – • A pair of angles that “sit on top of each other” when one vertex is put on top of the other. • Keys to this: • Do you see the word parallel in any of the definitions? • The angle pairs are one per vertex – meaning they are not adjacent, nor linear pair.

13. Transversals and Angle Pairs l 2 1 Alternate Interior 4 3 <3 & <6 5 6 <4 & <5 m 7 8 Alternate Exterior n <1 & <8 <2 & <7 Consecutive Exterior Consecutive Interior <1 & <7 <3 & <5 <2 & <8 <4 & <6

14. Transversals and Angle Pairs l Corresponding Angles 2 1 4 3 <1 & <5 5 6 m 7 8 <2 & <6 n <3 & <7 All the angle pairs have been made by the transversal, line n, intersecting two lines, l and m in two different places. <4 & <8

15. Chapter 3.2 Angles and Parallel Lines

16. Now, parallel lines! • We made a big deal last section that the angles are created by a transversal cutting two different lines in two different locations. • What happens when the transversal cuts parallel lines? • What happens is that the five named relationships we studied now have specific relationships.

17. Cutting Parallel Lines • When a transversal cuts parallel lines, the five pairs of angles have specific relationships. • Alternate Interior Angles (Alt Int) are congruent. • Alternate Exterior Angles (Alt Ext) are congruent. • Corresponding Angles (Corr) are congruent.

18. Cutting Parallel Lines (Con’t) • Consecutive Interior Angles (Con Int) are supplementary. • Consecutive Exterior Angles (Con Ext) are supplementary. • These relations hold true ONLY if the transversal cuts parallel lines. • If the lines are not parallel, then the relationships don’t exist.

19. Example < Pairs Trans Lines Name Equation 1 2 3 4 n 5 6 7 8 m||n 9 10 11 12 m 13 14 15 16 l p m, n Corr m<1 = m<9 <1 & <9 l n l, p Alt Int None <6 & <3 p m, n Alt Ext m<16 = m<3 <16 & <3 p m, n Con Ext <16 & <4 m<16+m<4=180

20. Perpendicular Transversals • Perpendicular Transversal Theorem – In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line.

21. Auxiliary Line • An Auxiliary Line is a line that you can draw any where you like. You can draw them parallel or perpendicular or not. 30° 30° Using the AAP, you get m<75° ? 45° 45°

22. Chapter 3.3 Slopes of Lines

23. Slope • The slope of a line is the ratio of the vertical rise to it’s horizontal run. • Slope = Rise Run • m = (y2 – y1)/(x2 – x1) • Slopes can tell you if two lines are parallel (||), perpendicular ( | ) or neither. • Slopes can not tell if two lines are congruent or not.

24. Slopes of Lines Create a right triangle Find vertical change 3 7 Find horizontal change m= 3/7

25. Four Types of Slopes Positive slope – line going “uphill” Negative slope – line going “downhill” Zero slope – there is “no hill” Undefined Slope – there is a “wall”

26. Parallel vs. Perpendicular • Slopes can tell you if a pair of lines are parallel, perpendicular or neither. • Parallel lines are lines that have the same slope. • If the slope of one line is 3/2 and the slope of the other line is also 3/2 then the two lines are parallel. • Slopes of zero are parallel. • Undefined slopes are parallel.

27. Parallel vs. Perpendicular • For two lines to be perpendicular the slopes have to be “opposite signed, reciprocals”. • This is pretty easy, if one line has a slope of m = 1/3 then the slope of the perpendicular line must be m = -3/1. • The only thing that is unusual is that if a line has a slope of zero then the slope of the perpendicular line is undefined. • The reverse is also true.

28. Chapter 3.4 Equations of Lines

29. Equations of Lines • There are three forms of a line equation that you should be familiar with. • Slope Intercept Form y = mx + b. • Standard Form Ax + By = C • Point Slope Form y – y1 = m(x – x1) • It doesn’t matter which form you use, you should be able to move from one form to the other simply.

30. Slope Intercept Form • Probably the most common form of a line equation is the Slope Intercept Form. • The reason why this is the most common form is once you have the line equation in this form you can simply read the slope and the y intercept. • You have y = mx + b, where m = slope of the line and b is the y intercept. • The coordinate of the y intercept is always (0, b)

31. Standard Form • The standard form of the line equation is Ax + By = C. • Both the x and y are on one side of the equal sign, while the constant is on the other side. • A, B and C are integers (no fractions or decimals) and A must be positive. • Here you can’t just read the slope and the y intercept, you will need to solve for them.

32. Point Slope Form • The point slope form may have been the first form a linear equation that you learned. • The equation is y – y1 = m(x – x1), where m is the slope and the point that the slope goes through is (x1, y1). • To use this equation you will need to plug in m for the slope and (x1, y1) the point.

33. Example – Slope Intercept FormGiven a point and slope • Find the equation of a line with a slope of 2/3 through the point (3, -4) • Substitute m= 2/3, x = 3 and y = -4 into the slope intercept form of the line equation. • You get -4 = (2/3)3 + b • Then solve for b. • b = -6 • Substitute m and b back into the equation. • y = (2/3)x - 6

34. Example – Slope Intercept FormGiven two points • Find the equation of a line through (-3, -1) and (6, 5) • First you need to find the slope. • m = (-1 – 5)/(-3 – 6) = -6/-9 = 2/3. • Now pick one point and do what we just did. • -1=(2/3)(-3) + b solving for b • b = 1 • So y=(2/3)x + 1

35. Another Example • Find the equation of the perpendicular line through the line connecting (3, 5) and ( 6, -2) through (-1, 4) • First, find the slope between the two points m=(-2 – 5)/(6 – 3) = -7/3 • Second, find the slope of the line perpendicular to the line with a slope of -7/3, so m = 3/7 • Next, solve for b: 4 = (3/7)(-1) + b, so b = 31/7 • So, y = (3/7)x + 31/7 is the equation of the perpendicular line.

36. Chapter 3.5 Proving Lines Parallel

37. Review from 3.2 • If a transversal cuts parallel lines, then… • alternate interior angle pairs are congruent. • alternate exterior angle pairs are congruent. • consecutive interior angle pairs are supplementary. • consecutive exterior angle pairs are supplementary. • corresponding angle pairs are congruent. • Now let us work on the converses of these statements.

38. Converses • If Alternate Interior Angle pairs are congruent… • If Alternate Exterior Angle pairs are congruent… • If Consecutive Interior Angle pairs are supplementary… • If Consecutive Exterior Angle pairs are supplementary… • If Corresponding Angle pairs are congruent.. • …then the transversal intersects parallel lines.

39. Examples m n 30° l n & p are || - Alt Int < Thrm 30° 150° p 30° l & m are || - Corr < Thrm 30° 30° n & p are || - Alt Ext < Thrm 30° l & m are || - Con Int < Thrm

40. Another Example Find the value of x to make these two lines parallel. 3x + 12° What kind of angles are these? Con Ext What do Con Ext anglesneed to be to make linesparallel? x + 5° (3x + 12) + (x + 5) = 180 Supplementary 4x + 17 = 180 4x = 163 x = 163/4

41. Chapter 3.6 Perpendiculars and Distance

42. Distance Between a Point and a Line • The distance between a point and a line you will need to find the length of the perpendicular segment from the point to that line.

43. A D C B Distance from a Point to a Line Find the distance between: B and line AD…. D and line AB D and line CB A and line CB B and line CD

44. Distance Between Parallel Lines • To find the distance between two parallel lines, just pick a point on one of the lines and find the distance between the point and a line. Alternate definition of Parallel Lines – Coplanar Lines that are Equidistant.

45. Parallel Line Theorem • In a plane, if two lines are equidistant from a third line, then the two lines are parallel to each other. • This is not an example of the transitive postulate.

46. Example – Coordinate Geo Find the distance fromC to line AB. Construct a segment perpendicular from C to line AB. Hint: If the line is horizontal or vertical, then just count the boxes. So, the distance between C and line AB is 6.

47. Example – Coordinate Geo (H) Find the distance from A to BC. Since BC is not Horor Vert, we need to do a lot of work. D First, find the slope of line BC. m = 3/1 Second, find equation of BC y = 3x + 13 Third, find the perpendicular slope. m = -1/3 Fourth, find the equation of the perpendicular line fromA to BC. y = (-1/3)x - 1

48. Example - Continued • So, now we have the equation of BC y = 3x + 13 & the equation of AD y = (-1/3)x – 1. • Since both equations are written in slope intercept form you can set them equal to each other. • 3x + 13 = (-1/3)x – 1….. mult. by 3 • 9x + 39 = -x – 3 • 10x = -42 x = - 4.2 and y = .4 • Now, find the distance between A ( 3, -2) and D ( - 4.2, 0.4)

49. Non-Euclidean Geometry (H) • Euclidean Geometry is what we are doing up to this date. It is the geometry of planes. • Non-Euclidean Geometry is the geometry of other types of systems such as spheres or cones….. • In Spherical Geometry, we are dealing with great circles, circles etc. • See page 165…