ACT Math Quiz: Triangles

20 questions · exam conditions

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TrianglesQuestion 1 of 20

In $\triangle XYZ$ , the measure of $\angle X$ is $40°$ and the measure of $\angle Y$ is $70°$ . What is the measure of $\angle Z$ ?

40°

70°

90°

110°

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ACT Math Quiz

ACT Math Quiz: Triangles

Practice Triangles in ACT Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Triangles, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

In $\triangle XYZ$ , the measure of $\angle X$ is $40°$ and the measure of $\angle Y$ is $70°$ . What is the measure of $\angle Z$ ?

$40°$
$70°$ (correct answer)
$90°$
$110°$

Explanation: The correct answer is B (70°). The interior angles of any triangle sum to 180°. Subtract the two known angles: 180° − 40° − 70° = 70°. A (40°) results from incorrectly assuming the triangle is isosceles and setting angle Z equal to angle X. C (90°) comes from assuming the triangle is a right triangle without justification. D (110°) is the most common error — adding the two known angles instead of subtracting their sum from 180°: 40 + 70 = 110. Pro tip: when you see 'find the third angle of a triangle,' immediately write 180 − (sum of the two given angles).

Question 2

In $\triangle ABC$ , $AB = 8$ , $AC = 12$ , and $\angle A = 30°$ . What is the area of $\triangle ABC$ ?

$24$ (correct answer)
$24\sqrt{3}$
$48$
$48\sqrt{3}$

Explanation: The correct answer is A (24). Use the trigonometric area formula: Area = (1/2) × a × b × sin(C), where a and b are two sides and C is the included angle. Area = (1/2)(8)(12) sin(30°) = (1/2)(96)(0.5) = 24. B (24√3) uses sin(60°) = √3/2 instead of sin(30°) = 1/2: (1/2)(96)(√3/2) = 24√3. C (48) forgets the 1/2 factor: (8)(12)(0.5) = 48. D (48√3) uses sin(60°) and omits the 1/2 factor: 96 × (√3/2) = 48√3. Pro tip: memorize sin(30°) = 1/2, sin(45°) = √2/2, and sin(60°) = √3/2. The 30° angle here makes the calculation clean.

Question 3

What type of triangle has one angle measuring more than 90°?

Acute
Obtuse (correct answer)
Right
Equilateral

Explanation: We need to identify the type of triangle with one angle measuring more than 90°. An obtuse triangle is defined as having exactly one angle greater than 90°. The other two angles must be acute (less than 90°) to maintain the 180° sum. Acute triangles have all angles less than 90°.

Question 4

In a right triangle, one leg measures 8 units and the hypotenuse is 10 units. What is the length of the other leg?

4 units
6 units (correct answer)
7 units
9 units

Explanation: We need to find the unknown leg of a right triangle with one leg = 8 units and hypotenuse = 10 units. Using the Pythagorean theorem: a² + b² = c². So 8² + b² = 10², which gives 64 + b² = 100, therefore b² = 36, and b = 6 units.

Question 5

In a 30-60-90 triangle, if the shortest side is 4 units, what is the length of the hypotenuse?

4√3 units
8 units (correct answer)
4 units
6 units

Explanation: We need to find the hypotenuse of a 30-60-90 triangle where the shortest side is 4 units. In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2, where the shortest side (opposite 30°) is 1. If the shortest side is 4, then the hypotenuse = 4 × 2 = 8 units.

Question 6

A 15-foot ladder is leaning against a vertical wall. The base of the ladder is 9 feet away from the base of the wall on level ground. How high up the wall, in feet, does the ladder reach?

6
12 (correct answer)
$\sqrt{135}$
24

Explanation: The correct answer is B (12). Apply the Pythagorean theorem: a² + b² = c², where the hypotenuse is 15 feet. 9² + b² = 15² → 81 + b² = 225 → b² = 144 → b = 12. This is the 3-4-5 Pythagorean triple scaled by 3: 9-12-15. A (6) comes from subtracting linearly: 15 − 9 = 6, treating the sides as lengths to subtract rather than using squares. C (√135) results from adding the squares instead of subtracting: 9² + 15² = 81 + 225 = 306... or a mismatch in which side is the hypotenuse. D (24) comes from adding: 9 + 15 = 24. Always identify the hypotenuse (longest side, opposite the right angle) before applying the theorem.

Question 7

In right triangle DEF, angle D is the right angle. If side DE = 3 and side DF = 4, what is the length of side EF?

5 (correct answer)
6
7
8

Explanation: We need to find the length of hypotenuse EF in right triangle DEF. Since D is the right angle, sides DE and DF are the legs, and EF is the hypotenuse. Using the Pythagorean theorem: DE² + DF² = EF², so 3² + 4² = EF². Calculating: 9 + 16 = 25, therefore EF = √25 = 5.

Question 8

In $\triangle ABC$ : $\angle A = 50°$ , $\angle B = 60°$ , $\angle C = 70°$ . What is the correct ordering of side lengths from least to greatest?

$AB < BC < AC$
$BC < AC < AB$ (correct answer)
$AC < BC < AB$
$BC < AB < AC$

Explanation: This is a triangle side-angle relationship question testing the fundamental theorem that larger angles are opposite longer sides. Choice B (BC < AC < AB) is correct — the shortest side is opposite the smallest angle, and the longest side is opposite the largest angle. Angle A = 50° (smallest) → opposite side BC is shortest. Angle B = 60° (middle) → opposite side AC is middle. Angle C = 70° (largest) → opposite side AB is longest. Order: BC < AC < AB. Choice A (AB < BC < AC) reverses the relationship entirely. Choice C (AC < BC < AB) correctly identifies AB as the longest but swaps BC and AC — reversing the two smaller sides. Choice D (BC < AB < AC) correctly identifies BC as shortest but places the other two in the wrong order. Pro tip: Always pair each angle with its opposite side: side BC is opposite angle A, side AC is opposite angle B, side AB is opposite angle C. Then rank the sides in the same order as their opposite angles. Draw a triangle and label if needed.

Question 9

In rectangle $WXYZ$ , the length is 15 cm and the diagonal is 17 cm. What is the width, in cm?

2
8 (correct answer)
16
$\sqrt{514}$

Explanation: This is a Pythagorean theorem question applied to a rectangle's diagonal. Choice B (8) is correct — the diagonal of a rectangle creates a right triangle with legs equal to the length and width. Pythagorean theorem: w² + 15² = 17² → w² + 225 = 289 → w² = 64 → w = 8. (Note: 8-15-17 is a Pythagorean triple.) Choice A (2) comes from subtracting linearly: 17 − 15 = 2 — using subtraction instead of the theorem. Choice C (16) doubles the correct answer — perhaps finding w² = 64, then computing 2w = 16 or misidentifying 64 as the side rather than the square. Choice D (√514) adds instead of subtracts in the theorem: w² = 15² + 17² = 225 + 289 = 514 — misidentifying which side is the hypotenuse. Pro tip: In a rectangle, the diagonal is always the hypotenuse (longest side). Subtract the squares of the known leg from the square of the hypotenuse to find the missing leg: w² = 17² − 15². Knowing common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) saves time.

Question 10

Triangle MNO is an equilateral triangle. What is the measure of angle M?

60° (correct answer)
90°
120°
45°

Explanation: We need to find the measure of angle M in equilateral triangle MNO. In an equilateral triangle, all three sides are equal in length and all three angles are equal in measure. Since the sum of angles in any triangle is 180°, each angle in an equilateral triangle measures 180° ÷ 3 = 60°.

Question 11

In right triangle XYZ, angle X is the right angle. If side XY = 6 and side XZ = 8, what is the length of side YZ?

10 (correct answer)
9
11
13

Explanation: We need to find the length of hypotenuse YZ in right triangle XYZ. Since X is the right angle, sides XY and XZ are the legs, and YZ is the hypotenuse. Using the Pythagorean theorem: XY² + XZ² = YZ², so 6² + 8² = YZ². Calculating: 36 + 64 = 100, therefore YZ = √100 = 10.

Question 12

Triangle STU is a 45-45-90 triangle. If the hypotenuse ST is 10, what is the length of each leg?

10
5
$10\sqrt{2}$
$5\sqrt{2}$ (correct answer)

Explanation: We need to find the length of each leg in a 45-45-90 triangle with hypotenuse ST = 10. In a 45-45-90 triangle, the sides are in the ratio 1 : 1 :

\sqrt{2}

, where the legs are equal and the hypotenuse is

\sqrt{2}

times the leg length. If hypotenuse = 10, then leg

\times \sqrt{2}

= 10, so leg = 10 /

\sqrt{2}

= 10

\sqrt{2}

/ 2 = 5

\sqrt{2}

Question 13

In triangle DEF, angle D measures 80° and angle E measures 70°. What is the measure of angle F?

20°
30° (correct answer)
40°
60°

Explanation: We need to find the measure of angle F in triangle DEF. Using the triangle angle sum theorem, all three angles must sum to 180°. Setting up the equation: 80° + 70° + F = 180°, so 150° + F = 180°. Solving for F: F = 180° - 150° = 30°.

Question 14

Triangle GHI is a 45-45-90 triangle with legs measuring 5 units each. What is the length of the hypotenuse?

5 units
10 units
7 units
5√2 units (correct answer)

Explanation: We need to find the hypotenuse of a 45-45-90 triangle with legs of 5 units each. In a 45-45-90 triangle, the hypotenuse equals leg × √2. With legs of 5 units, the hypotenuse = 5√2 units. Choice B (10 units) would apply the incorrect formula of 2 × leg instead of leg × √2.

Question 15

In triangle DEF, angle D measures 85°, angle E measures 45°. What is the measure of angle F?

80°
60°
70°
50° (correct answer)

Explanation: We need to find angle F in triangle DEF where angle D = 85° and angle E = 45°. Using the triangle angle sum theorem, all angles sum to 180°. So angle F = 180° - 85° - 45° = 50°. This triangle is a scalene triangle with all different angles.

Question 16

What type of triangle has all angles equal?

Isosceles
Equilateral (correct answer)
Scalene
Right

Explanation: We need to identify the type of triangle where all angles are equal. In an equilateral triangle, all three sides are equal and all three angles are equal, each measuring 60°. This is a fundamental property that distinguishes equilateral triangles from other types. Isosceles triangles have only two equal angles.

Question 17

In the grayscale diagram, triangle $ABC$ is right with a right angle at $B$ . The side lengths are labeled $AB=5$ , $BC=12$ , and $AC=x$ . What is the length of side $x$ ?

(Use $a^2+b^2=c^2$ with $c$ as the hypotenuse.)

$\sqrt{169}$
$13$ (correct answer)
$\sqrt{119}$
$17$

Explanation: We need to find the hypotenuse x in right triangle ABC where AB = 5 and BC = 12, with the right angle at B. Using the Pythagorean theorem:

AB^2 + BC^2 = AC^2

. Calculating:

5^2 + 12^2 = 25 + 144 = 169

, so

x^2 = 169

and

x = 13

. Choice A shows

\sqrt{169}

, which equals 13 but is not simplified.

Question 18

If two angles of a triangle measure $38^\circ$ and $90^\circ$ , what is the measure of the third angle?

$142^\circ$
$128^\circ$
$232^\circ$
$52^\circ$ (correct answer)

Explanation: We need to find the third angle when two angles measure 38° and 90°. The sum of angles in any triangle is 180°. Setting up the equation: 38° + 90° + x = 180°, so 128° + x = 180°, which gives x = 52°. Choice B shows 128°, which is the sum of the two given angles, not the third angle.

Question 19

In triangle GHI, angle G measures 45° and angle H measures 45°. What type of triangle is GHI?

Right
Equilateral
Scalene
Isosceles (correct answer)

Explanation: We need to classify triangle GHI where angles G and H both measure 45°. Using the triangle angle sum theorem: 45° + 45° + I = 180°, so angle I = 90°. Since the triangle has two equal angles (45° each) and two equal sides opposite those angles, it is an isosceles triangle. It's also a right triangle since one angle is 90°.

Question 20

In triangle VWX, angle V measures 30°, and angle W measures 60°. What is the measure of angle X?

120°
60°
30°
90° (correct answer)

Explanation: We need to find the measure of angle X in triangle VWX. Using the triangle angle sum theorem, all three angles must sum to 180°. Setting up the equation: 30° + 60° + X = 180°, so 90° + X = 180°. Solving for X: X = 180° - 90° = 90°.