2.2: Parallel Lines and Transversals | Geometry Honors

How it works

When parallel lines are cut by a transversal, corresponding and alternate interior angles are congruent; same-side interior angles are supplementary. The converses can prove lines parallel.

\ell\parallel m\ \Rightarrow\ \alpha=\beta\quad\text{(corresponding angles)}

Worked examples

Example 1

Corresponding angles on parallel lines measure $\displaystyle (5x+5)^\circ$ and $\displaystyle 80^\circ$ . Find $\displaystyle x$ .

$\displaystyle 5x+5=80$ , so $\displaystyle x=15$ .

Example 2

Same-side interior angles are $\displaystyle 110^\circ$ and $\displaystyle y^\circ$ on parallel lines. Find $\displaystyle y$ .

They sum to $\displaystyle 180^\circ$ , so $\displaystyle y=70^\circ$ .

Problems

Problem 2.2.1

Parallel lines are cut by a transversal. One acute angle is $\displaystyle 60^\circ$ . Find the other acute angles and the obtuse angles.

Two horizontal parallel lines cut by a rising transversal; one acute angle is 60 degrees. — The two horizontal lines are given parallel.

Show worked solution

All acute angles are $\displaystyle 60^\circ$ by corresponding and vertical angle relationships. All obtuse angles are $\displaystyle 120^\circ$ by linear pairs.

Two horizontal parallel lines cut by a rising transversal; one acute angle is 60 degrees. The solution adds labeled relationships. — Corresponding acute angles are 60 degrees; supplementary obtuse angles are 120 degrees.

Problem 2.2.2

Alternate interior angles measure $\displaystyle (3x+12)^\circ$ and $\displaystyle (5x-28)^\circ$ on parallel lines. Find their measure.

Show worked solution

Set $\displaystyle 3x+12=5x-28$ . Then $\displaystyle x=20$ and each angle is $\displaystyle 72^\circ$ .

Problem 2.2.3

Can equal corresponding angles prove two lines parallel?

Show worked solution

Yes. The converse of the corresponding angles theorem applies to two coplanar lines cut by a transversal.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 2.2.1

Problem 2.2.2

Problem 2.2.3

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