Lesson 3-5 Proving Lines Parallel

• Postulate 3.4- If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.

Example:

• Postulate 3.5- Parallel PostulateIf a given line and a point not on the line, then

there exists exactly one line through the point that is parallel to the given line.

Proving Lines Parallel

Theorems Examples3.5 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

3.6 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.

3.7 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

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

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:

Determine which lines, if any, are parallel.

Answer:

ALGEBRA Find x and mZYN so that

Explore From the figure, you know that

and You also know that are alternate exterior

angles.

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

Answer:

Original equation

Simplify.

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

ALGEBRA Find x and mGBA so that

Answer:

Given:

Prove:

Proof:

1. Given1.

5. Substitution5.

ReasonsStatements

2. Consecutive Interior Thm.

2. . 3. Def. of suppl. s3.

4. Def. of congruent s4.

6. Def. of suppl. s6. . 7. If cons. int. s are suppl.,

then lines are .7.

Given:

Prove:

Proof:

1. Given1. 2. Alternate Interior Angles2.

3. Substitution3.

4. Definition of suppl. s4. .

5. Definition of suppl. s5.

6. Substitution6.

7. If cons. int. s are suppl., then lines are .

7.

ReasonsStatements

Answer:

Answer: Since the slopes are not equal, r is not parallel to s.