SAT
Topic 2
Triangles & similarity
Angle sum, similar triangles, and special right-triangle ratios.
Concept
Foundational triangle facts every SAT problem uses:
- Angle Sum Theorem — interior angles always sum to 180°.
- Exterior Angle Theorem — an exterior angle equals the sum of the two non-adjacent (remote) interior angles.
- Similar triangles have equal corresponding angles and proportional corresponding sides. If the linear ratio is a:b, the area ratio is a²:b².
Special right triangles — memorise these ratios. The SAT loves them:
- 30-60-90 — sides 1 : √3 : 2 opposite the 30°, 60°, 90° angles.
- 45-45-90 — sides 1 : 1 : √2 (an isosceles right triangle).
Worked example 1
A triangle has angles 50° and 70°. Find the third angle.
Solution
Sum. Three angles total 180°.
Compute. 180 − 50 − 70 = 60°
Third angle: 60°.
Worked example 2
A 30-60-90 triangle has hypotenuse 8. Find the lengths of the two legs.
Solution
Ratios. Sides are in ratio 1 : √3 : 2. The hypotenuse is the largest, length 2k.
Solve. 2k = 8 ⇒ k = 4.
Legs. Short leg (opposite 30°) = k = 4. Long leg (opposite 60°) = k√3 = 4√3.
Legs: 4 and 4√3 ≈ 6.93.
Practice test
8 questions on angle sum, similarity, and special right triangles.
Practice test · 8 questions
Question 1 of 8 · Score 0