Geometry and Trigonometry · Math

Lines, angles, and triangles

Lines, angles, and triangles: angle sums, parallel lines cut by a transversal, similar triangles, and exterior angles.

19 questions in the bank · 5 easier · 8 medium · 6 harder · this domain is ≈15% of the section

Learn it in the Guide

What to watch for

With parallel lines, decide whether your pair is congruent or supplementary before computing. For similar triangles, match the correspondence of vertices before writing any ratio.

How the question is worded

These are the actual stems used in the bank — the SAT reuses a small set of wordings, and recognising them saves time on test day.

  • In a triangle, the measures of two angles are 39° and 31°. What is the measure, in degrees, of the third angle?
  • In a triangle, the measures of two angles are 73° and 70°. What is the measure, in degrees, of the third angle?
  • In a triangle, the measures of two angles are 37° and 42°. What is the measure, in degrees, of the third angle?
  • In a triangle, the measures of two angles are 38° and 65°. What is the measure, in degrees, of the third angle?

Worked examples

easyIn a triangle, the measures of two angles are 39° and 31°. What is the measure…

In a triangle, the measures of two angles are 39° and 31°. What is the measure, in degrees, of the third angle?

  • A. 20
  • B. 70
  • C. 110
  • D. 141

Answer: C. 110

Explanation

The three angle measures of any triangle sum to 180°: 180 − 39 − 31 = 110.

mediumLines m and n are parallel and are intersected by a transversal, forming alter…

Lines m and n are parallel and are intersected by a transversal, forming alternate angles with measures x° and 57°. What is the value of x?

  • A. 33
  • B. 57
  • C. 123
  • D. 303

Answer: B. 57

Explanation

With parallel lines, alternate angles are congruent when the lines are parallel. So x = 57.

hardTriangle ABC is similar to triangle DEF, where A, B, and C correspond to D, E,…

Triangle ABC is similar to triangle DEF, where A, B, and C correspond to D, E, and F. Side AB has length 8, side BC has length 12, and side DE has length 24. What is the length of side EF?

Answer: 36

Explanation

Corresponding sides of similar triangles share one ratio. DE/AB = 24/8 = 3, so every side of DEF is 3 times its partner: EF = 3 · 12 = 36. Matching the correspondence (AB↔DE, BC↔EF) before writing the ratio is the whole game.