SAT · Math · Geometry and Trigonometry

Lines, angles, and triangles

Lines, angles, and triangles is part of Geometry and Trigonometry in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include parallel lines cut by a transversal, triangle angle sum and exterior angles, similar triangles and side ratios and isosceles and equilateral facts.

Common traps: setting supplementary angles equal; setting corresponding angles to sum to 180; finding x but reporting x instead of the angle.

Sample questions

Question 1
A land surveyor measures the shadow of a 2-metre marker post as 3 metres. At the same moment, a tall pylon casts a shadow of 12 metres on level ground. Assuming similar triangles, how tall is the pylon, in metres?
Enter your own answer.
Question 2
Triangle ABC is similar to triangle DEF, with AB corresponding to DE and BC corresponding to EF. AB = 4, DE = 6 and BC = 9. What is the length of EF?
- A.
  6
- B.
  9
- C.
  11
- D.
  13.5
Answer: D. 13.5
The scale factor from ABC to DEF is 6/4 = 1.5, so EF = 1.5 × 9 = 13.5, choice D. Choice A applies the inverse ratio 4/6, B copies BC, and C adds the 2-unit difference between AB and DE, instead of multiplying.
Question 3
Triangle ABC is similar to triangle DEF, with A, B, and C corresponding to D, E, and F. AB = 10, BC = 3, and DE = 40. What is the length of EF?
- A.
  3/4
- B.
  12
- C.
  33
- D.
  120
Answer: B. 12
The scale factor is DE/AB = 40/10 = 4. So EF = 3 × 4 = 12.
Question 4
In triangles ABC and DEF, AB = DE and angle B = angle E. Which additional piece of information is enough to prove that triangle ABC is congruent to triangle DEF?
- A.
  BC = EF
- B.
  AC = DF
- C.
  BC = DF
- D.
  angle A + angle C = angle D + angle F
Answer: A. BC = EF
BC = EF gives two sides and the included angle B = E, so SAS proves congruence (A). B gives SSA, since angle B is not between AB and AC, which does not guarantee congruence; C pairs sides that do not correspond; D follows automatically from angle B = angle E and adds nothing.
Question 5
In triangle PQR, angle P measures (2x + 5)° and angle Q measures (x + 25)°. The exterior angle formed by extending side PR beyond R measures (5x + 10)°. What is the measure, in degrees, of angle PRQ?
- A.
  25
- B.
  60
- C.
  120
- D.
  145
Answer: C. 120
The exterior angle equals the sum of the two remote interior angles: 5x + 10 = 3x + 30, so x = 10 and P = 25°, Q = 35°. Then angle PRQ = 180 − 60 = 120° (C). Choice A is angle P; B is the exterior angle itself; D is 180 − Q.

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