Chapter 3-5Chapter 3-5

Proving Lines ParallelProving Lines Parallel

Lesson 3-5 Ideas/Vocabulary

• Recognize angle conditions that occur with parallel lines.• Prove that two lines are parallel based on given angle

relationships.

Standard 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key)Standard 16.0 Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line. (Key)

Transitive property of Parallels• If two lines are parallel to the same line, then

they are parallel to each other.• If p // q and q // r, then p // r.

p

q

r

Reminders from Section 1We will use these same theorems to

prove the lines are parallel given certain angle information.

Corresponding Angle TheoremIf two parallel lines are cut by a transversal, then corresponding

angles are congruent.

// lines corresponding s are

Corresponding Angle Theorem

Alternate Interior Angle TheoremIf two parallel lines are cut by a

transversal, then alternate interior angles are congruent.

// lines Alt. Int. s are

Alternate Interior Angle Theorem

Alternate Exterior Angle TheoremIf two parallel lines are cut by a

transversal, then alternate exterior angles are congruent.

// lines Alt. Ext. s are

Alternate Exterior Angle Theorem

Consecutive Interior Angle TheoremIf two parallel lines are cut by a

transversal, then consecutive interior angles are supplementary.

// lines Consec. Int. s are Supp.

Consecutive Interior Angle Theorem

1

2

m1 + m2 = 180

Two Theorem• 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.

p

m

n

Lesson 3-5 Postulates

Animation: Construct a Parallel Line Through a Point not on Line

Lesson 3-5 Theorems

Lesson 3-5 Example 1

Determine which lines, if any, are parallel.

Identify Parallel Lines

77oConsec. Int. s are supp.

a//bAlt. Int. s are not

a is not // c

Consec. Int. s are not supp.

b is not // c

A B

C D

0% 0%0%0%

Lesson 3-5 CYP 1

A. A

B. B

C. C

D. D

I only

II only

III only

I, II, and III

Determine which lines, if any are parallel.I. e || fII. e || gIII. f || g

ALGEBRA Find x and m ZYN so that || .

Lesson 3-5 Example 2

Solve Problems with Parallel Lines

Explore From the figure, you know that m WXP = 11x – 25 and m ZYN = 7x + 35. You also know that WXP and ZYN are alternate exterior angles.

Lesson 3-5 Example 2

If Alt. Ext. angles are , then the lines will be // m WXP = m ZYN Alternate exterior thm.11x – 25 = 7x + 35 Substitution4x – 25 = 35 Subtract 7x from each side.

4x = 60 Add 25 to each side.x = 15 Divide each side by 4.

ALGEBRA Find x and m ZYN so that || .

Lesson 3-5 Example 2

Solve Problems with Parallel Lines

Now use the value of x to find m ZYN.

Answer: x = 15, m ZYN = 140

m ZYN = 7x + 35 Original equation= 7(15) + 35 x = 15= 140 Simplify.

A B

C D

0% 0%0%0%

Lesson 3-5 CYP 2

A. A

B. B

C. C

D. D

ALGEBRA Find x so that || .x = 60

x = 9

x = 12

x = 12

Lesson 3-5 Example 3

Prove Lines Parallel

Prove: r || s

Given: ℓ || m

Lesson 3-5 Example 3

Prove Lines Parallel

2. 2. Consecutive Interior Angle Theorem

5. 5. Substitution 6. 6. Definition of supplementary

angles7. 7. If consecutive interior angles

theorem

1. 1. Given

Proof:Statements Reasons

4. 4. Definition of congruent angles

3. 3. Definition of supplementary

angles

Given x || y and , can you use theCorresponding Angles Postulate to prove a || b?

A. A

B. B

C. C

A B

C

0% 0%0%

Lesson 3-5 CYP 3

yes

no

not enough informationto determine

Lesson 3-5 Example 4

Determine whether p || q.

Slope and Parallel Lines

slope of p:

slope of q:

Answer: Since the slopes are equal, p || q.

A B

C

0% 0%0%

A. A

B. B

C. C

Lesson 3-5 CYP 4

Yes, r is parallel to s.

No, r is not parallel to s.

It cannot be determined.

Determine whether r || s.

HomeworkChapter 3-5• Pg 175

1 – 5, 7 – 19, 23 (proof), 24(proof), 37, 50 – 52