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
In the xy-plane, lines ℓ and m are parallel and are both intersected by a third line. One of the angles formed at the intersection with ℓ measures (3x + 10)°, and the corresponding angle at the intersection with m measures (5x - 30)°. What is the value of x?
- A.
  5
- B.
  10
- C.
  15
- D.
  20
Answer: D. 20
Corresponding angles formed by a transversal of parallel lines are equal, so 3x + 10 = 5x - 30. Subtracting 3x and adding 30 gives 40 = 2x, so x = 20; both angles then measure 70°. A drops the -30, B subtracts 10 from 30 instead of adding (2x = 20), and C drops the +10.
Question 2
Parallel lines p and q are cut by a transversal t. One angle formed by t and p measures (3x + 10)°, and the corresponding angle formed by t and q measures (5x − 30)°. What is the measure, in degrees, of each of these angles?
- A.
  20
- B.
  25
- C.
  70
- D.
  110
Answer: C. 70
Corresponding angles are equal, so 3x + 10 = 5x − 30, giving x = 20 and an angle of 3(20) + 10 = 70° (C). A reports x instead of the angle; B sets the angles supplementary (8x − 20 = 180); D is the supplement of the correct angle.
Question 3
A triangle has two sides of lengths 6 and 11. Which of the following could be the length of the third side?
- A.
  4
- B.
  5
- C.
  13
- D.
  17
Answer: C. 13
The third side must be greater than 11 − 6 = 5 and less than 11 + 6 = 17, and only 13 lies in that range (C). A is too short to close the triangle; B and D equal the boundary values, which would flatten the triangle into a segment.
Question 4
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 5
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.

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