DWG NO. 03.1 — Lesson 1 of 4

Transversals and Angle Relationships

Unit 3: Parallel and Perpendicular Lines · ~15–30 min

Objective

Identify the angle pairs formed when a transversal crosses two lines, and use the parallel-line angle theorems to find unknown angle measures.

A transversal and its angle pairs

A transversal is a line that crosses two (or more) other lines at two distinct points. Where it crosses each line, it creates four angles — eight angles total — and those eight angles fall into four named relationships depending on where they sit relative to the two lines and the transversal.

Pair type	Where they sit	If the two lines are parallel
Corresponding	Same position at each intersection (both upper–left, both lower–right, etc.)	Congruent
Alternate interior	Between the two lines, on opposite sides of the transversal	Congruent
Alternate exterior	Outside the two lines, on opposite sides of the transversal	Congruent
Consecutive interior	Between the two lines, on the same side of the transversal	Supplementary

Transversal t crosses lines m and n at P and Q, forming angles 1–4 at P and 5–8 at Q.

The parallel-line angle theorems

These relationships only guarantee congruence or supplementary angles when the two lines cut by the transversal are parallel. That condition matters — it's the hinge the next lesson turns on.

Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.
Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.

Worked Example 1 · Finding every angle

ProblemIn the diagram above, line m ∥ line n and angle 3 = 115°. Find the measures of angles 1, 2, 4, 5, 6, 7, and 8.

1.At point P, angle 3 and angle 1 form a linear pair (they're supplementary): angle 1 = 180° − 115° = 65°. Angle 4 is vertical to angle 1, so angle 4 = 65°. Angle 2 is vertical to angle 3, so angle 2 = 115°.

2.Angle 3 and angle 6 are alternate interior angles, so by the Alternate Interior Angles Theorem, angle 6 = 115°. Angle 4 and angle 5 are also alternate interior angles, so angle 5 = 65°.

3.At point Q, angle 7 is vertical to angle 6, so angle 7 = 115°. Angle 8 is vertical to angle 5, so angle 8 = 65°.

Angle 1 = 65° • Angle 2 = 115° • Angle 4 = 65° • Angle 5 = 65° • Angle 6 = 115° • Angle 7 = 115° • Angle 8 = 65°

Worked Example 2 · Solving for x

ProblemLine m ∥ line n. Angle 2 = (3x + 15)° and angle 6 = (5x − 25)°. Find x and the measure of angle 2.

1.Angle 2 and angle 6 occupy the same position at their intersections (both upper–right), so they are corresponding angles — and the lines are parallel, so they're congruent.

2.Set the expressions equal: 3x + 15 = 5x − 25.

3.Solve: 40 = 2x, so x = 20. Then angle 2 = 3(20) + 15 = 75°.

x = 20 • Angle 2 = 75°

Guided practice

1.Using the diagram above, name the angle that is consecutive interior with angle 3.
Show answer
Angle 5. Angles 3 and 5 both lie between lines m and n, on the same side of transversal t.
2.Line m ∥ line n and angle 4 = 72°. Find angle 6, and name the theorem that justifies your answer.
Show answer
Angle 6 = 108°. Angle 4 and angle 6 are consecutive interior angles, so by the Consecutive Interior Angles Theorem they're supplementary: 180° − 72° = 108°.
3.Line m ∥ line n and angle 1 = 58°. Find angle 8.
Show answer
58°. Angle 1 and angle 8 are alternate exterior angles, which are congruent when the lines are parallel.
4.Angle 2 = (4x)° and angle 3 = (2x + 30)°. These two angles are a linear pair at point P. Find x.
Show answer
x = 25. Linear pairs are supplementary: 4x + 2x + 30 = 180, so 6x = 150 and x = 25.
5.True or false: corresponding angles are always congruent, even if the two lines cut by the transversal are not parallel. Explain.
Show answer
False. The Corresponding Angles Postulate only guarantees congruence when the two lines are parallel — if they aren't, corresponding angles can have any relationship.

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