3.5 Proving Lines Parallel3.5 Proving Lines Parallel

Objectives Recognize angle conditions that occur with

parallel lines

Prove that two lines are parallel based on given angle relationships

Postulate 3.4Converse of the Corresponding Angles Postulate

If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

Abbreviation: If corr. s are , then lines are ║.

Postulate 3.5 Parallel Postulate

If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line.

Theorem 3.5 Converse of the Alternate Exterior Angles Theorem

If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel.

Abbreviation: If alt ext. s are , then lines are ║.

Theorem 3.6Converse of the Consecutive Interior Angles Theorem

If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel.

Abbreviation: If cons. int. s are supp., then lines are ║.

Proof of the Converse of the Consecutive Interior Angles Theorem

Given: 4 and 5 are supplementary

Prove: g ║ h

6 g

h

5

4

Paragraph Proof of the Converse of the Consecutive Interior Angles Theorem

You are given that 4 and 5 are supplementary. By the Supplement Theorem, 5 and 6 are also supplementary because they form a linear pair. If 2 s are supplementary to the same , then 4 6. Therefore, by the Converse of the Corresponding s Angles Postulate, g and h are parallel.

Theorem 3.7Converse of the Alternate Interior Angles Theorem

If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.

Abbreviation: If alt. int. s are , then lines are ║.

Theorem 3.8

In a plane, if two lines are perpendicular to the same line, then they are parallel.

Abbreviation: If 2 lines are ┴ to the same line, then the lines are ║.

Determine which lines, if any, are parallel.

consecutive interior angles are supplementary. So,

consecutive interior angles are not supplementary. So, c is not parallel to a or b.Answer:

Example 1:Example 1:

Determine which lines, if any, are parallel.

Answer:

Your Turn:Your Turn:

ALGEBRA Find x and mZYN so that

Explore From the figure, you know that and You also know that

are alternate exterior angles.

Example 2:Example 2:

Alternate exterior angles

Subtract 7x from each side.

Substitution

Add 25 to each side.

Divide each side by 4.

Solve

Plan For line PQ to be parallel to MN, the alternate exterior angles must be congruent. Substitute the given angle measures into this equation and solve for x. Once you know the value of x, use substitution to find

Example 2:Example 2:

Answer:

Original equation

Simplify.

Examine Verify the angle measure by using the value of x to find

Since

Example 2:Example 2:

ALGEBRA Find x and mGBA so that

Answer:

Your Turn:Your Turn:

Assignment

Geometry: Pg. 155 #13 – 31, 34, 35

Pre-AP Geometry: Pg. 155 #13 – 32, 34 - 40