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Pre-AP Bellwork. 1) Solve for p. (3p – 5)°. 3-2 Proving Lines Parallel. Postulate 3-2: Converse of the Corresponding Angles Postulate. If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

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Pre ap bellwork
Pre-AP Bellwork

1) Solve for p.

(3p – 5)°



Postulate 3 2 converse of the corresponding angles postulate
Postulate 3-2: Converse of the Corresponding Angles Postulate

  • If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.


Theorem 3 3 converse of the alternate interior angles theorem
Theorem 3-3: Converse of the Alternate Interior Angles Theorem

  • If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.


Theorem 3 4 converse of the sam side interior angles theorem
Theorem 3-4: Converse of the Sam-Side Interior Angles Theorem

  • If two lines and a transversal form same-side interior angles that are supplementary, then the lines are parallel.


Theorem 3 10 alternate exterior angles converse
Theorem 3.10: Alternate Exterior Angles Converse Theorem

  • 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
Prove the Alternate Interior Angles Converse Theorem

Given: 1  2

Prove: m ║ n

3

m

2

1

n


Example 1 proof of alternate interior converse

Statements: Theorem

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
Proof of the Consecutive Interior Angles Converse Theorem

Given: 4 and 5 are supplementary

Prove: g ║ h

g

6

5

4

h


Paragraph proof
Paragraph Proof Theorem

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.


Find the value of x that makes j k

Solution: Theorem

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
Using Parallel Converses: TheoremUsing 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
Solution: Theorem

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
Example 5: Identifying parallel lines Theorem

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 lines1
Example 5: Identifying parallel lines Theorem

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 lines2
Example 5: Identifying parallel lines Theorem

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
Conclusion: Theorem

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.