Determining which diagram shows two parallel lines intersected by a transversal is a common geometry question. It tests knowledge of the special angle relationships that result when parallel lines are cut by a transversal. This guide provides a step-by-step method to identify parallel lines cut by a transversal in a diagram using angle theorems and reasoning.

## Introduction to Parallel Lines and Transversals

In geometry, parallel lines are two lines in the same plane that never intersect and maintain the same distance between them. A transversal is a third line that intersects two other lines at distinct points.

When a transversal cuts through parallel lines, it forms specific pairs of congruent (equal measure) angles including:

- Corresponding angles – Matching angles on the same side of the transversal and on the same side of the parallel lines.
- Alternate interior angles – The angles between the parallel lines and on opposite sides of the transversal.
- Alternate exterior angles – Outside the parallel lines and on opposite sides of the transversal.
- Same-side interior angles – Inside the parallel lines and on the same side of the transversal. These angles are supplementary (add up to 180°).

## Step-by-Step Method to Identify Parallel Lines

When trying to determine which diagram shows parallel lines cut by a transversal, follow these steps:

**Step 1)** Look for a diagram with two lines crossed by a third line (the transversal).

**Step 2)** Identify the 8 angles formed by the intersection points. Label each angle if needed.

**Step 3)** Determine if any angles are already proven congruent based on their angle measure markings.

**Step 4)** Apply theorems about angles formed by parallel lines cut by a transversal:

- If a pair of corresponding angles are congruent, the lines are parallel.
- If a pair of alternate interior angles are congruent, the lines are parallel.
- If a pair of alternate exterior angles are congruent, the lines are parallel.
- If same-side interior angles are supplementary (add to 180°), the lines are parallel.

**Step 5)** If one of the above angle conditions is met, conclude the diagram shows parallel lines cut by a transversal.

## Examples

**Question:** Which diagram shows parallel lines cut by a transversal?

**Diagram 1:**

[Diagram shows two lines intersected by a third line. Angles A and F are marked 115°.]

**Diagram 2:**

[Diagram shows two lines intersected by a third line. Angles B and E are marked 105°.]

**Solution:** Diagram 1 shows parallel lines cut by a transversal because the corresponding angles A and F are congruent (both 115°). Since corresponding angles formed by the transversal are congruent, the two crossed lines must be parallel.

**Question:** In the diagram below, lines 1 and 2 are cut by transversal t. Does this show parallel lines?

[Diagram shows two lines cut by a transversal. Angles A and D are marked x°. Angle B is marked y°.]

**Solution:** Yes, this is a diagram of parallel lines cut by a transversal t. Angles A and D are alternate exterior angles and their measures are equal (both x°). The Alternate Exterior Angles Theorem states that if two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel. Since angles A and D are congruent alternate exterior angles, lines 1 and 2 must be parallel.

## Recognizing Non-Parallel Lines Cut by a Transversal

To prove lines are not parallel when cut by a transversal, look for:

- No pairs of congruent corresponding, alternate interior, or alternate exterior angles
- Same-side interior angles that are not supplementary
- Known angle measures that do not obey parallel line angle theorems
- geometric clues like intersecting lines or non-flat angles

If none of the angle conditions indicating parallel lines are met, then the lines cut by the transversal are likely not parallel.

## Conclusion

Analyzing angles is the key to determining if a diagram shows two parallel lines intersected by a transversal. Look for congruent corresponding, alternate interior, or alternate exterior angles. Or same-side interior angles that are supplementary. If any of these angle relationships are present, you can conclude the diagram exhibits parallel lines cut by a transversal. Mastering this important geometry skill helps solve many types of parallel line problems.