Chapter 3

Perpendicular and Parallel Lines

Chapter Objectives

Identify parallel linesDefine angle relationships between parallel linesDevelop a Flow ProofUse Alternate Interior, Alternate Exterior, Corresponding, & Consecutive Interior AnglesCalculate slopes of Parallel and Perpendicular lines

Lesson 3.1

Lines and Angles

Lesson 3.1 Objectives

Identify relationships between lines.Identify angle pairs formed by a transversal.Compare parallel and skew lines.

Lines and Angle Pairs

1

5

2

8

4

6

3

7Corresponding Angles – because they lie in corresponding positions of each intersection.

TransversalAlternate Exterior Angles – because they lie outside the two lines and on opposite sides of the transversal.

Alternate Interior Angles – because they lie inside the two lines and on opposite sides of the transversal.

Consecutive Interior Angles – because they lie inside the two lines and on the same side of the transversal.

Example 1

Determine the relationship between the given angles1) 3 and 9

1) Alternate Interior Angles

2) 13 and 52) Corresponding Angles

3) 4 and 103) Alternate Interior Angles

4) 5 and 154) Alternate Exterior Angles

5) 7 and 145) Consecutive Interior Angles

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.

Example 2

Complete the following statements using the words parallel, skew, perpendicular.

1) Line WZ and line XY are _________.1) parallel

2) Line WZ and line QW are ________.2) perpendicular

3) Line SY and line WX are _________.3) skew

4) Plane WQR and plane SYT are _________.4) parallel

5) Plane RQT and plane WQR are _________.5) perpendicular

6) Line TS and line ZY are __________.6) skew

7) Line WX and plane SYZ are __________.7) parallel.

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.

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.

Homework 3.1

In Class 2-9

p132-135

Homework 10-31

Due Tomorrow

Lesson 3.2

Proof and Perpendicular Lines

Lesson 3.2 Objectives

Develop a Flow ProofProve results about perpendicular linesUse Algebra to find angle measure

Flow Proof

A flow proof uses arrows to show the flow of a logical argument.

Each reason is written below the statement it justifies.

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

PROVE: 5 7

GIVEN: 5 and 6 are a linear pair 6 and 7 are a linear pair

56 7

5 and 6 are a linear pair

6 and 7 are a linear pair

5 and 6 are supplementary

6 and 7 are supplementary

5 7

Given Given

Linear PairPostulate

Linear PairPostulate

Congruent Supplements Theorem

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

Theorem 3.2:Adjacent Angles Complementary

If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

Theorem 3.3: Four Right Angles

If two lines are perpendicular, then they intersect to form four right angles.

Homework 3.2

None!Move on to Lesson 3.3

Lesson 3.3

Parallel Lines and Transversals

Lesson 3.3 Objectives

Prove lines are parallel using tranversals.Identify properties of parallel lines.

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

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

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

31 2

8 7

+=180o + = 180o

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

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!

Example 3 Find the missing angles for the following:

120o

120o

105o

105o

110o

110o 70o

120o

60o

140o

140o

Homework 3.3

In Class 3-6

p146-149

Homework 8-26, 34-44 even

Due TomorrowQuiz Wednesday Lessons 3.1-3.3

Emphasis on 3.1 & 3.3

Lesson 3.4

Proving Lines are Parallel

Lesson 3.4 Objectives

Prove that lines are parallelRecall the use of converse statements

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

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

31 2

8 7

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

31 2

8 7

+=180o + = 180o

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

Example 4Is it possible to prove the lines are parallel?

If so, explain how.

Yes they are parallel!

Because Corresponding Angles are congruent.

Corresponding Angles Converse

Yes they are parallel!

Because Alternate Interior Angles are congruent.

Alternate Interior Angles Converse

Yes they are parallel!

Because Alternate Exterior Angles are congruent.

Alternate Exterior Angles Converse

No they are not parallel!

No relationship between those two angles.

Example 5Find the value of x that makes the m n.

x = 2x – 95 (AIA)

-x = –95 (SPOE)

x = 95 (DPOE)

100 = 4x – 28 (CA)

128 = 4x (APOE)

x = 32 (DPOE)

(3x + 15) + 75 = 180(CIA)

3x + 90 = 180 (CLT)

3x = 90 (SPOE)

x = 30 (DPOE)

Directions do not ask for reasons, I am showing you them because I am a teacher!!

Homework 3.4

In Class 1, 3-9

p153-156

Homework 10-35, 37, 38

Due Tomorrow

Lesson 3.5

Using Properties of Parallel Lines

Lesson 3.5 Objectives

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

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.

p q r

If p // q and q // r, then p // r.

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.

m n

p

If m p and n p, then m // n.

Finding Parallel Lines

Find any lines that are parallel and explain why.

a b c

x

y

z

125o

55o

Results

x // y 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

Homework 3.5

In Class 16, 19

p160-163

Homework 8-24, 33-36, 43-51

Due Tomorrow

Lesson 3.6

Parallel Lines in the Coordinate Plane

Lesson 3.6 Objectives

Review the slope of a lineIdentify parallel lines based on their slopesWrite equations of parallel lines in a coordinate plane

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.

Remember that slope is often referred to asrise

run

Which really meansy2 – y1

x2 – x1

Which we find it in an equation by looking for m. y = mx + b

= m

Example of SlopeYou are given two points:

Now label each point as 1 and 2.1 2

Then substitute as the formula for slope tells you.

3 )B( , 81 )A( , 2

8=

– 2

3 – 1

y1y2– x2– x1

=6

2= 3

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 = -1 m = undefined

Writing an Equation inSlope Intercept Form

You will be given Slope

Or at least two points so you can calculate slope

y-intercept y = mx + b

slope y-intercept

This is the point at which the line touches the y-axis.

Your final answer should always appear in this form.

Writing an Equation GivenSlope 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 passesTo 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 algebraWhen 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.

Example

Example 5 p167

Write an equation of the line through the point (2,3) that has a slope of 5.

y = mx + b

y = 5x + b

3 = 5(2) + b

3 = 10 + b

b = -7

So, your final answer is:

y = 5x - 7

Homework 3.6

WSDue Tomorrow

Lesson 3.7

Perpendicular Lines in the Coordinate Plane

Lesson 3.7 Objectives

Use slope to identify perpendicular linesWrite equations of perpendicular lines

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.

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

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!

Tougher Example

Example 4 P173Decide whether the lines are perpendicular

Line r: 4x + 5y = 2 Line s: 5x + 4y = 3

Need to change into slope-intercept form.

y = mx + b

Subtract x-term from both sides5y = -4x + 2 4y = -5x + 3

Get y to be alone by dividing off the coefficienty = -4/5x + 2/5 y = -5/4x + 3/4

Now multiply their slopes

(-4/5)(-5/4) =20/20 = 1

NOT Perpendicular

Homework 3.7

WSDue TomorrowTest Thursday January 29th