Lines & Angles • Topic 2 of 3

Parallel Lines & Transversals

When a straight line (a transversal) crosses two parallel lines it creates eight angles, but only two distinct values appear. Corresponding angles (same position at each intersection) are equal, alternate interior angles (between the parallels, opposite sides of the transversal) are equal, and alternate exterior angles are equal. The one pair that is NOT equal is co-interior (same-side interior) angles: they are supplementary, adding to 180°. A reliable shortcut is that every one of the eight angles is either equal to your reference angle or supplementary to it — so each angle is one of just two numbers that themselves add to 180°. The GRE often marks the lines parallel with matching arrowheads; use those marks, not the drawing, since it is not to scale.

Two parallel lines cut by a transversal, with equal alternate interior angles markedParallel lines cut by a transversalAlternate interior (and corresponding) angles are equal

✅ Solved examples

1. Two parallel lines are cut by a transversal. One angle measures 65°. Find its corresponding angle.
Corresponding angles are equal, so the corresponding angle is also 65°.
2. Two parallel lines are cut by a transversal. A co-interior (same-side interior) angle is 110°. Find the other co-interior angle.
Co-interior angles are supplementary: 180° − 110° = 70°.
3. Alternate interior angles are (3x − 5)° and (2x + 20)°. Find x and the angle measure.
Alternate interior angles are equal: 3x − 5 = 2x + 20 → x = 25. Angle = 2(25) + 20 = 70°.
4. A transversal cuts two parallel lines. One interior angle is x° and the co-interior angle on the same side is (x + 40)°. Find x.
Same-side interior angles are supplementary: x + (x + 40) = 180 → 2x + 40 = 180 → 2x = 140 → x = 70.

✏️ Practice — try these, take hints as needed

1. Two parallel lines are cut by a transversal; one angle is 48°. Find its corresponding angle.
Corresponding angles are equal.
No calculation needed.
The value is unchanged.
48°
2. A same-side interior angle measures 123°. Find the other same-side interior angle.
Co-interior angles are supplementary.
They sum to 180°.
180 − 123.
57°
3. Alternate exterior angles are (4x)° and (2x + 50)°. Find the angle measure.
Alternate exterior angles are equal.
4x = 2x + 50.
x = 25, then 4x.
100°
4. Co-interior angles are (2x + 10)° and (3x)°. Find x.
They are supplementary.
(2x + 10) + 3x = 180.
5x + 10 = 180.
x = 34
5. Two parallel lines are cut by a transversal. One interior angle is 5 times its co-interior partner. Find the smaller angle.
Let the smaller be a; the other is 5a.
They are supplementary: a + 5a = 180.
6a = 180.
30°

📝 Topic test — 8 questions

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