Lines, Angles, Triangles

Unit 4 · Geometry & Trigonometry

What the SAT asks here

The SAT tests three angle skills: vertical/linear-pair/parallel-line relationships, the triangle interior angle sum (and the exterior-angle shortcut), and similar-triangle proportions. Expect 1–2 problems per test.

Angle relationships

Vertical angles are equal

Linear pair sums to 180°

Parallel lines + transversal — corresponding angles equal, alternate interior angles equal

Triangle facts

Interior angles sum to 180°

Exterior angle = sum of the two non-adjacent interior angles

Isosceles — equal sides give equal opposite angles

Equilateral — all sides equal, all angles 60°

Similar triangles

Corresponding angles equal; corresponding sides in constant ratio

AA postulate: two equal angles → similar

SAT Tip: When the problem mentions parallel lines, look for transversal angle pairs first — they often unlock the rest of the figure in one step.

Caution: Triangles can look similar in a figure without actually being similar. The SAT expects you to justify similarity from given information, not appearance.

Type 1

Angle relationships

Apply vertical-angle, linear-pair, or parallel-line/transversal facts to set an equation, then solve for the unknown.

Example 1

Lines ℓ and m are parallel, cut by a transversal. One pair of alternate interior angles measures (2x + 10)° and (3x − 20)°. Find x.

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Type 2

Triangle angle sum

Use the 180° total inside a triangle, or the exterior-angle rule when one of the angles is outside the triangle.

Example 2

A triangle has angles (x)°, (2x)°, (3x)°. Find the smallest angle.

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Type 3

Similar triangles

Set up a proportion between corresponding sides. AA (two pairs of equal angles) is the SAT's go-to justification for similarity.

Example 3

Triangle ABC is similar to triangle DEF with ratio 2 : 5. If AB = 8, find the corresponding side DE.

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