ACT · Math · Preparing for Higher Math: Geometry

Angles, triangles, and similarity

Angles, triangles, and similarity is part of Preparing for Higher Math: Geometry in the ACT 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 triangle angle sum and exterior angles, parallel lines and a transversal, similar triangles and proportion and isosceles and equilateral properties.

Common traps: using 360 degrees; treating the exterior angle as an interior angle; solving for x but not the angle.

Sample questions

Question 1
A floor tile is shaped like a hexagon. What is the sum of the measures of its interior angles?
- A.
  360°
- B.
  540°
- C.
  720°
- D.
  1,080°
Answer: C. 720°
The interior angles of an n-sided polygon sum to (n − 2) · 180°. For a hexagon, (6 − 2) · 180° = 720°. Choice A is the sum of the exterior angles; B uses n − 3, the sum for a pentagon; D uses n · 180°.
Question 2
The angles of a triangular sail measure x°, 2x° and (x + 20)°. What is the measure of the largest angle of the sail?
- A.
  40°
- B.
  60°
- C.
  80°
- D.
  170°
Answer: C. 80°
The angles of a triangle sum to 180°: 4x + 20 = 180, so x = 40 and the angles are 40°, 80° and 60°. The largest is 80°. Choice A is x itself; B is the angle x + 20; D uses 360° as the angle sum (x = 85, so 2x = 170).
Question 3
In triangle PQR, point S is on side PQ and point T is on side PR so that segment ST is parallel to side QR. PS = 4, SQ = 6 and ST = 5. What is the length of QR?
- A.
  2
- B.
  7.5
- C.
  11
- D.
  12.5
Answer: D. 12.5
Since ST ∥ QR, triangle PST is similar to triangle PQR with ratio PS/PQ = 4/10. So 5/QR = 4/10 and QR = 12.5. Choice A multiplies by the ratio instead of dividing (5 · 4/10); B compares the parts SQ and PS instead of using the whole side PQ; C adds 6 to 5 as if the lengths grew by the same amount.
Question 4
Two similar triangular sails have areas of 18 square meters and 50 square meters. What is the ratio of the perimeter of the smaller sail to the perimeter of the larger sail?
- A.
  81/625
- B.
  9/25
- C.
  3/5
- D.
  5/3
Answer: C. 3/5
The area ratio is 18/50 = 9/25, which is the square of the scale factor, so the ratio of lengths, and of perimeters, is √(9/25) = 3/5. Choice A squares the area ratio; B uses the area ratio for perimeters; D inverts the ratio, giving larger to smaller.
Question 5
In triangle ABC, AB = 12, AC = 18 and BC = 20. The bisector of angle A meets side BC at point D. What is the length of BD?
- A.
  8
- B.
  10
- C.
  12
- D.
  40/3
Answer: A. 8
An angle bisector divides the opposite side in the ratio of the adjacent sides: BD/DC = AB/AC = 12/18 = 2/3, so BD = (2/5) · 20 = 8. Choice B treats the bisector as a median; C is the length of DC; D multiplies 20 by 12/18 instead of splitting 20 in the ratio 2 : 3.

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