SAT · ACT · MCATGoal-based prepLive 1:1 + GoalGap AI
ACT · Math · Preparing for Higher Math: Geometry
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.
Question 1
One angle of a parallelogram-shaped floor tile measures 68°. What is the measure of an angle of the tile that is consecutive with (next to) the 68° angle?
22°
68°
112°
292°
Answer: C. 112°
Consecutive angles of a parallelogram are supplementary, so the adjacent angle is 180° − 68° = 112°. Choice A is the complement of 68°; B is the opposite angle, which is equal to 68°, not the consecutive one; D subtracts from 360°.
Question 2
In an aquarium shop, two triangular glass braces, △ABC and △DEF, are checked. It is known that AB = DE, AC = DF, and ∠A = ∠D. Which statement justifies that the braces are congruent?
SSS: all three pairs of sides are equal.
SAS: two pairs of sides and the angle between them are equal.
AAA: all three pairs of angles are equal.
SSA: two sides and a non-included angle are equal.
Answer: B. SAS: two pairs of sides and the angle between them are equal.
Angle A lies between sides AB and AC (and D between DE and DF), so SAS applies. A would need BC = EF, which is not given; C AAA gives similar triangles, not necessarily congruent; D SSA is not a valid criterion and misdescribes the angle, which is between the two sides.
Question 3
Lines ℓ and m are parallel and are cut by a transversal. Two angles on the same side of the transversal and between the parallel lines measure (3x + 10)° and (2x + 20)°. What is the measure, in degrees, of the larger of the two angles?
30
40
80
100
Answer: D. 100
Same-side interior angles are supplementary: 5x + 30 = 180, so x = 30. The angles are 100° and 80°; the larger is 100°. Choice A (30) gives the value of x instead of an angle. Choice B (40) sets the angles equal, giving x = 10. Choice C (80) gives the smaller angle.
Question 4
Two sides of a triangle measure 7 cm and 12 cm. How many whole-number lengths are possible for the third side?
12
13
14
15
Answer: B. 13
B is right: The third side x satisfies 12 − 7 < x < 12 + 7, so 5 < x < 19. The whole numbers 6 through 18 give 13 values. A counts 6 to 17, off by one. C includes the length 5 or 19. D includes both 5 and 19.
Question 5
In a triangular training loop, the second angle is 20° larger than the first angle, and the third angle is 3 times the first angle. What is the measure, in degrees, of the smallest angle?
32
36
52
96
Answer: A. 32
Let the first angle be x. Then x + (x + 20) + 3x = 180, so 5x = 160 and x = 32. The angles are 32°, 52°, and 96°. Choice B (36) divides 180 by 5 and ignores the extra 20°. Choice C (52) is the second angle. Choice D (96) is the largest angle, 3 times the first.
A few questions show the style. The free scan shows which ACT skills cost you the most points, then builds a plan around them.