1. Perpendicular and Parallel Lines
Chapter 3
Perpendicular and Parallel Lines

2. Chapter Objectives Identify parallel lines
Define angle relationships between parallel lines
Develop a Flow Proof
Use Alternate Interior, Alternate Exterior, Corresponding, & Consecutive Interior Angles
Calculate slopes of Parallel and Perpendicular lines

3. Lesson 3.1
Lines and Angles

4. Lesson 3.1 Objectives Identify relationships between lines.
Identify angle pairs formed by a transversal.
Compare parallel and skew lines.

5. Lines and Angle Pairs
Alternate Exterior Angles – because they lie outside the two lines and on opposite sides of the transversal.
Transversal 2 1 3 4
Consecutive Interior Angles – because they lie inside the two lines and on the same side of the transversal. 5 6
Corresponding Angles – because they lie in corresponding positions of each intersection. 8 7
Alternate Interior Angles – because they lie inside the two lines and on opposite sides of the transversal.

6. Example 1 Determine the relationship between the given angles
3 and 9
Alternate Interior Angles
13 and 5
Corresponding Angles
4 and 10
5 and 15
Alternate Exterior Angles
7 and 14
Consecutive Interior Angles

7. Parallel versus Skew
Two lines are parallel if they are coplanar and do not intersect.
Lines that are not coplanar and do not intersect are called skew lines.
These are lines that look like they intersect but do not lie on the same piece of paper.
Skew lines go in different directions while parallel lines go in the same direction.

8. Example 2
Complete the following statements using the words parallel, skew, perpendicular.
Line WZ and line XY are _________.
parallel
Line WZ and line QW are ________.
perpendicular
Line SY and line WX are _________.
skew
Plane WQR and plane SYT are _________.
Plane RQT and plane WQR are _________.
Line TS and line ZY are __________.
Line WX and plane SYZ are __________.
parallel.

9. Parallel and Perpendicular Postulates: Postulate 13-Parallel Postulate
If there is a line and a point not on the line, then there is exactly one line through the point that is parallel to the given line.

10. Parallel and Perpendicular Postulates: Postulate 14-Perpendicular Postulate
If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.

11. Homework 3.1
In Class
2-9 p
Homework
10-31
Due Tomorrow

12. Proof and Perpendicular Lines
Lesson 3.2
Proof and Perpendicular Lines

13. Lesson 3.2 Objectives Develop a Flow Proof
Prove results about perpendicular lines
Use Algebra to find angle measure

14. Flow Proof 5 6 7
A flow proof uses arrows to show the flow of a logical argument.
Each reason is written below the statement it justifies.
GIVEN:  5 and  6 are a linear pair
 6 and  7 are a linear pair
5 and  6 are a linear pair
 6 and  7 are a linear pair
PROVE:  5   7
Given
Given
1.  5 and  6 are a linear pair
 6 and  7 are a linear pair
1. Given
2.  5 and  6 are supplementary
 6 and  7 are supplementary
2. Linear Pair Postulate
3.  5   7
3. Congruent Supplements Theorem
 5 and  6 are supplementary
 6 and  7 are supplementary
Linear Pair
Postulate
Linear Pair
Postulate
 5   7
Congruent Supplements Theorem

15. Theorem 3.1: Congruent Angles of a Linear Pair
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. g h
So g  h

16. Theorem 3.2: Adjacent Angles Complementary
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

17. Theorem 3.3: Four Right Angles
If two lines are perpendicular, then they intersect to form four right angles.

18. Homework 3.2
None!
Move on to Lesson 3.3

19. Parallel Lines and Transversals
Lesson 3.3
Parallel Lines and Transversals

20. Lesson 3.3 Objectives Prove lines are parallel using tranversals.
Identify properties of parallel lines.

21. Postulate 15: Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
You must know the lines are parallel in order to assume the angles are congruent. 1 5 2 8 4 6 3 7

22. Theorem 3.4: Alternate Interior Angles
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Again, you must know that the lines are parallel.
If you know the two lines are parallel, then identify where the alternate interior angles are.
Once you identify them, they should look congruent and they are. 1 5 2 8 4 6 3 7

23. Theorem 3.5: Consecutive Interior Angles
If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.
Again be sure that the lines are parallel.
They don’t look to be congruent, so they MUST be supplementary. 5 4 6 3 1 2 8 7
180o = + + =
180o

24. Theorem 3.6: Alternate Exterior Angles
If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.
Again be sure that the lines are parallel. 1 2 8 5 4 6 3 7

25. Theorem 3.7: Perpendicular Transversal
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Again you must know the lines are parallel.
That also means that you now have 8 right angles!

26. Example 3 Find the missing angles for the following: 120o 120o 120o

27. Homework 3.3 In Class Homework Due Tomorrow Quiz Wednesday 3-6p
Homework
8-26, even
Due Tomorrow
Quiz Wednesday
Lessons
Emphasis on 3.1 & 3.3

28. Proving Lines are Parallel
Lesson 3.4
Proving Lines are Parallel

29. Lesson 3.4 Objectives Prove that lines are parallel
Recall the use of converse statements

30. Postulate 16: Corresponding Angles Converse
If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
You must know the corresponding angles are congruent.
It does not have to be all of them, just one pair to make the lines parallel. 1 5 2 8 4 6 3 7

31. Theorem 3.8: Alternate Interior Angles Converse
If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.
Again, you must know that alternate interior angles are congruent. 5 4 6 3 1 2 8 7

32. Theorem 3.9: Consecutive Interior Angles Converse
If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.
Be sure that the consecutive interior angles are supplementary. 5 4 6 3 1 2 8 7
180o = + + =
180o

33. Theorem 3.10: Alternate Exterior Angles Converse
If two lines are cut by a transversal so that alternate exterior angles are congruent, then lines are parallel.
Again be sure that the alternate exterior angles are congruent. 1 2 8 5 4 6 3 7

34. Example 4 Is it possible to prove the lines are parallel?
If so, explain how.
Yes they are parallel!
Yes they are parallel!
Because Alternate Interior Angles are congruent.
Because Corresponding Angles are congruent.
Corresponding Angles Converse
Alternate Interior Angles Converse
Yes they are parallel!
Because Alternate Exterior Angles are congruent.
No they are not parallel!
Alternate Exterior Angles Converse
No relationship between those two angles.

35. Example 5 Find the value of x that makes the m  n.
x = 2x – 95 (AIA)‏
100 = 4x – 28 (CA)‏
(3x + 15) + 75 = 180(CIA)‏
-x = –95 (SPOE)‏
128 = 4x (APOE)‏
3x + 90 = 180 (CLT)‏
x = 95 (DPOE)‏
x = 32 (DPOE)‏
3x = 90 (SPOE)‏
x = 30 (DPOE)‏
Directions do not ask for reasons, I am showing you them because I am a teacher!!

36. Homework 3.4 In Class Homework Due Tomorrow 1, 3-9 10-35, 37, 38p
Homework
10-35, 37, 38
Due Tomorrow

37. Using Properties of Parallel Lines
Lesson 3.5
Using Properties of Parallel Lines

38. Lesson 3.5 Objectives
Prove more than two lines are parallel to each other.
Identify all possible parallel lines in a figure.

39. Theorem 3.11: 3 Parallel Lines Theorem
If two lines are parallel to the same line, then they are parallel to each other.
This looks like the transitive property for parallel lines.
If p // q and q // r, then p // r. p q r

40. Theorem 3.12: Parallel Perpendicular Lines Theorem
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
If m  p and n  p, then m // n. m n p

41. Finding Parallel Lines
Find any lines that are parallel and explain why. a b c x y z
125o
55o

42. Results x // y y // z x // z b // c Corresponding Angles Converse
Postulate 16
y // z
Consecutive Interior Angels Converse
Theorem 3.9
x // z
3 Parallel Lines Theorem
Theorem 3.11
b // c
Alternate Exterior Angles Converse
Theorem 3.10

43. Homework 3.5 In Class Homework Due Tomorrow 16, 19 8-24, 33-36, 43-51p
Homework
8-24, 33-36, 43-51
Due Tomorrow

44. Parallel Lines in the Coordinate Plane
Lesson 3.6
Parallel Lines in the Coordinate Plane

45. Lesson 3.6 Objectives Review the slope of a line
Identify parallel lines based on their slopes
Write equations of parallel lines in a coordinate plane

46. Slope
Recall that slope of a nonvertical line is a ratio of the vertical change divided by the horizontal change.
It is a measure of how steep a line is.
The larger the slope, the steeper the line is.
Slope can be negative or positive whether or not the lines slants up or down.
rise
Remember that slope is often referred to as
run
y2 – y1
Which really means
= m
x2 – x1
y = mx + b
Which we find it in an equation by looking for m.

47. 3 Example of Slope – x2 x1 – – 1 2 y1 y2 You are given two points: ), 1 2 ) B( , 3 8
Now label each point as 1 and 2. 1 2
Then substitute as the formula for slope tells you. y1 y2 – x2 x1 – 8 2 6 3 = = = – 3 1 2

48. Postulate 17: Slopes of Parallel Lines Postulate
In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope.
Any two vertical lines are parallel.
m = -1
m = undefined
m = -1

49. Writing an Equation in Slope Intercept Form
You will be given
Slope
Or at least two points so you can calculate slope
y-intercept
Your final answer should always appear in this form.
y = mx + b
y-intercept
slope
This is the point at which the line touches the y-axis.

50. Writing an Equation Given Slope and 1 Point
For this, you will be given the slope
Or have to determine it from and equation
Or determine it from a set of two points
You will also be given 1 point through which the line passes
To solve, use your slope-intercept form to find b
y = mx+b
Plug in
Slope for m.
The x-value from your point for x.
The y-value from your point for y.
Solve for b using algebra
When finished, be sure to rewrite in slope intercept form using your new m and b.
Leave x and y as x and y in your final equation.

51. Example y = 5x - 7 So, your final answer is: Example 5 y = mx + b
Write an equation of the line through the point (2,3) that has a slope of 5.
y = mx + b
So, your final answer is:
y = 5x + b
3 = 5(2) + b
y = 5x - 7
3 = 10 + b
b = -7

52. Homework 3.6 WS
Due Tomorrow

53. Perpendicular Lines in the Coordinate Plane
Lesson 3.7
Perpendicular Lines in the Coordinate Plane

54. Lesson 3.7 Objectives Use slope to identify perpendicular lines
Write equations of perpendicular lines

55. Postulate 18: Slopes of Perpendicular Lines Postulate
In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is –1.
Vertical and horizontal lines are perpendicular.

56. Identifying Perpendicular Lines
There are two ways to identify perpendicular lines
The product of the slopes equal –1
(3/2)(-2/3) = -1
Or to get from one slope to the other, you find the negative reciprocal
Remember that reciprocal flips the number.
Well now you flip it and make it negative!
3/2  -2/3

57. Determining Perpendicular Lines
Remember there are two ways to identify perpendicular lines
Multiply the slopes together
If the answer is –1, they are perpendicular
Verify that the slopes are negative reciprocals of each other
Take one of the slopes, flip it over, and make it negative. If the answer matches the other slope, they are perpendicular.
Example (#25 p176)
y = 3x
y = -1/3x – 2
(3)(-1/3) = -1
Perpendicular
Or 3  1/3  -1/3
Check!

58. Tougher Example y = mx + b Example 4 Line r: 4x + 5y = 2
Need to change into slope-intercept form.
y = mx + b
Example 4
P173
Decide whether the lines are perpendicular
Line r: 4x + 5y = 2
Line s: 5x + 4y = 3
Subtract x-term from both sides
5y = -4x + 2
4y = -5x + 3
Get y to be alone by dividing off the coefficient
y = -4/5x + 2/5
y = -5/4x + 3/4
Now multiply their slopes
NOT Perpendicular
(-4/5)(-5/4) =
20/20
= 1

59. Homework 3.7 WS
Due Tomorrow
Test Thursday
January 29th