SAT-67: Parallel Lines Cut by a Transversal

SAT-67: Parallel Lines Cut by a Transversal

Description: When a third line (a transversal) crosses two parallel lines, eight angles appear — but only two different sizes. Knowing the angle pairs lets you find them all from a single known angle. This is one of the most common SAT geometry setups.

The big idea: only two angle sizes

With parallel lines, every angle is either equal to your known angle or supplementary to it (adds to 180°). So once you know one angle, you know all eight. (Oʻzbekcha: parallel chiziqlarda barcha burchaklar yo teng, yo 180° ga toʻldiruvchi boʻladi.)

The angle pairs to recognize

• Corresponding angles (same position at each intersection) → equal.
• Alternate interior angles (between the lines, opposite sides of the transversal) → equal.
• Alternate exterior angles (outside the lines, opposite sides) → equal.
• Co-interior / same-side interior angles (between the lines, same side) → supplementary (180°).

(Oʻzbekcha: mos burchaklar va almashinuvchi ichki burchaklar teng; bir tomonli ichki burchaklar 180° ga teng.)

Worked Example 1 — corresponding angles

A transversal crosses two parallel lines. One angle is 75°. Find its corresponding angle at the other line.

• Corresponding angles are equal → the matching angle is also 75°.

Worked Example 2 — same-side interior

One interior angle is 120°. Find the same-side interior angle (between the lines, same side of the transversal).

• Same-side interior angles are supplementary: 180 − 120 = 60°.

(Oʻzbekcha: bir tomonli ichki burchaklar yigʻindisi 180° ga teng.)

Worked Example 3 — solve for x

Two parallel lines are cut by a transversal. Alternate interior angles are (2x + 10)° and (3x − 20)°. Find x and the angle.

• Alternate interior angles are equal: 2x + 10 = 3x − 20 → 30 = x → x = 30.
• Angle = 2(30) + 10 = 70° (check: 3(30) − 20 = 70 ✓).

Shortcut: in these figures every angle is one of just two values, A or 180 − A. Find which group each angle is in. (Oʻzbekcha: har bir burchak yoki A, yoki 180 − A boʻladi — qaysi guruhdaligini aniqlang.)

Worked Example 4 — supplementary pair to solve x

Same-side interior angles are (3x)° and (x + 40)°. Find x.

• Same-side interior angles add to 180: 3x + (x + 40) = 180 → 4x + 40 = 180 → 4x = 140 → x = 35.

(Oʻzbekcha: bir tomonli ichki burchaklar 180° ga teng, shuning uchun ularni qoʻshib tenglama tuzamiz.)

How to tell "equal" from "supplementary" fast

You don't have to memorize every pair name under pressure. Use this quick test: if the two angles look about the same size, they are equal; if one looks sharp (acute) and the other wide (obtuse), they are supplementary and add to 180°. The SAT figures are drawn roughly to scale, so this visual check almost always confirms which rule to use — then back it up with the pair name. (Oʻzbekcha: ikki burchak bir xil koʻrinsa — teng; biri oʻtkir, biri keng koʻrinsa — yigʻindisi 180°.)

The converse (proving lines are parallel)

The relationships also run backwards: if corresponding angles are equal (or same-side interior angles add to 180°), then the two lines must be parallel. The SAT sometimes gives the angles and asks whether the lines are parallel — check the pair. (Oʻzbekcha: agar mos burchaklar teng boʻlsa, chiziqlar parallel boʻladi.)

Practice 1

A transversal crosses two parallel lines; one angle is 65°. What is its alternate exterior angle?

Practice 2

Corresponding angles are (4x)° and (x + 60)°. Find x.

Key words — Kalit soʻzlar

• Parallel lines — parallel chiziqlar
• Transversal — kesuvchi chiziq
• Corresponding angles — mos burchaklar
• Alternate interior — almashinuvchi ichki
• Alternate exterior — almashinuvchi tashqi
• Same-side interior — bir tomonli ichki
• Interior / Exterior — ichki / tashqi
• Equal — teng
• Supplementary — yigʻindisi 180°

Summary

• A transversal across parallel lines makes only two angle sizes.
• Corresponding, alternate interior, and alternate exterior angles are equal.
• Same-side interior angles are supplementary (180°).