3.4 Proving Lines are Parallel

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3.4 Proving Lines are Parallel. Mrs. Spitz Fall 2005. Standard/Objectives:. Standard 3: Students will learn and apply geometric concepts Objectives: Prove that two lines are parallel.

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### 3.4 Proving Lines are Parallel

Mrs. Spitz

Fall 2005

Standard/Objectives:

Standard 3: Students will learn and apply geometric concepts

Objectives:

• Prove that two lines are parallel.
• Use properties of parallel lines to solve real-life problems, such as proving that prehistoric mounds are parallel.
HW ASSIGNMENT:
• 3.4--pp. 153-154 #1-28

Quiz after section 3.5

Postulate 16: Corresponding Angles Converse
• If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
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.
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.
Theorem 3.10: Alternate Exterior Angles Converse
• If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.
Prove the Alternate Interior Angles Converse

Given: 1  2

Prove: m ║ n

3

m

2

1

n

Statements:

1  2

2  3

1  3

m ║ n

Reasons:

Given

Vertical Angles

Transitive prop.

Corresponding angles converse

Example 1: Proof of Alternate Interior Converse
Proof of the Consecutive Interior Angles Converse

Given: 4 and 5 are supplementary

Prove: g ║ h

g

6

5

4

h

Paragraph Proof

You are given that 4 and 5 are supplementary. By the Linear Pair Postulate, 5 and 6 are also supplementary because they form a linear pair. By the Congruent Supplements Theorem, it follows that 4  6. Therefore, by the Alternate Interior Angles Converse, g and h are parallel.

Solution:

Lines j and k will be parallel if the marked angles are supplementary.

x + 4x = 180 

5x = 180 

X = 36 

4x = 144 

So, if x = 36, then j ║ k.

Find the value of x that makes j ║ k.

4x

x

Using Parallel Converses:Using Corresponding Angles Converse

SAILING. If two boats sail at a 45 angle to the wind as shown, and the wind is constant, will their paths ever cross? Explain

Solution:

Because corresponding angles are congruent, the boats’ paths are parallel. Parallel lines do not intersect, so the boats’ paths will not cross.

Example 5: Identifying parallel lines

Decide which rays are parallel.

H

E

G

58

61

62

59

C

A

B

D

A. Is EB parallel to HD?

B. Is EA parallel to HC?

Example 5: Identifying parallel lines

Decide which rays are parallel.

H

E

G

58

61

B

D

• Is EB parallel to HD?

mBEH = 58

m DHG = 61 The angles are corresponding, but not congruent, so EB and HD are not parallel.

Example 5: Identifying parallel lines

Decide which rays are parallel.

H

E

G

120

120

C

A

• B. Is EA parallel to HC?

m AEH = 62 + 58

m CHG = 59 + 61

AEH and CHG are congruent corresponding angles, so EA ║HC.

Conclusion:

Two lines are cut by a transversal. How can you prove the lines are parallel?

Show that either a pair of alternate interior angles, or a pair of corresponding angles, or a pair of alternate exterior angles is congruent, or show that a pair of consecutive interior angles is supplementary.