Parallel Lines Cut by a Transversal

What you'll learn

Identify corresponding, alternate interior, and alternate exterior angle pairs
Use angle relationships to find unknown angle measures
Prove that lines are parallel from angle information

Words to know

Transversal —: a line that crosses two or more other lines
Corresponding angles —: matching angles in the same position at each crossing
Alternate interior angles —: angles between the lines on opposite sides of the transversal
Supplementary —: two angles whose measures add to 180 degrees
Parallel —: lines in a plane that never meet

Why it matters The angle rules for parallel lines let builders, designers, and mapmakers guarantee that beams, streets, and tracks stay perfectly parallel.

Worked examples

Model ItI do

Lines m and n are parallel and cut by transversal t. If angle 2 measures 110 degrees, find angle 3.

Angle 2 and angle 3 are alternate interior angles (between the lines, on opposite sides of t). When the lines are parallel, alternate interior angles are equal. So angle 3 = 110 degrees.

Try It TogetherWe do

Using the same figure, angle 1 and angle 2 form a straight line at m. If angle 1 = 70 degrees, find angle 2.

Angle 1 and angle 2 are a linear pair, so they are supplementary: angle 1 + angle 2 = 180 degrees. Then angle 2 = 180 - 70 = 110 degrees.

Now YouYou do

Two parallel lines are cut by a transversal. A pair of corresponding angles measures 3x and 75 degrees. Find x.

Corresponding angles are equal, so set 3x = 75. Solve for x and check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

1.Corresponding angles at two parallel lines cut by a transversal are always _____.
2.One angle of a linear pair measures 130 degrees. Find the other angle.
3.Alternate interior angles at parallel lines lie between the lines on which sides of the transversal?

On Level

The core Quick Check from the book.

1.Corresponding angles at parallel lines are always _____.
2.If alternate interior angles measure 4x and 60, and the lines are parallel, find x.
3.Two same-side interior angles at parallel lines measure 65 and y. Find y.
4.If two alternate exterior angles are equal, what can you conclude about the two lines?
5.One angle of a linear pair is 128 degrees. Find the other.

Challenge

Stretch your thinking.

1.Two parallel lines are cut by a transversal. A pair of alternate interior angles measure 5x + 10 and 3x + 40. Find x.
2.Two same-side interior angles at parallel lines measure 2x and 3x. Find x.
3.Lines are cut by a transversal. A pair of corresponding angles measure 4x + 5 and 2x + 35. For the lines to be parallel, find x.