ACT Math Quiz: Triangles

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.

Question 1

1. $40°$
2. $70°$ (correct answer)
3. $90°$
4. $110°$

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

1. $24$ (correct answer)
2. $24\sqrt{3}$
3. $48$
4. $48\sqrt{3}$

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

1. Acute
2. Obtuse (correct answer)
3. Right
4. 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

1. 4 units
2. 6 units (correct answer)
3. 7 units
4. 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

1. 4√3 units
2. 8 units (correct answer)
3. 4 units
4. 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

1. 6
2. 12 (correct answer)
3. $\sqrt{135}$
4. 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

1. 5 (correct answer)
2. 6
3. 7
4. 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

1. $AB < BC < AC$
2. $BC < AC < AB$ (correct answer)
3. $AC < BC < AB$
4. $BC < AB < AC$

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

1. 2
2. 8 (correct answer)
3. 16
4. $\sqrt{514}$

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

1. 60° (correct answer)
2. 90°
3. 120°
4. 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

1. 10 (correct answer)
2. 9
3. 11
4. 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

1. 10
2. 5
3. $10\sqrt{2}$
4. $5\sqrt{2}$ (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

1. 20°
2. 30° (correct answer)
3. 40°
4. 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

1. 5 units
2. 10 units
3. 7 units
4. 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

1. 80°
2. 60°
3. 70°
4. 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

1. Isosceles
2. Equilateral (correct answer)
3. Scalene
4. 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

1. $\sqrt{169}$
2. $13$ (correct answer)
3. $\sqrt{119}$
4. $17$

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

1. $142^\circ$
2. $128^\circ$
3. $232^\circ$
4. $52^\circ$ (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

1. Right
2. Equilateral
3. Scalene
4. 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

1. 120°
2. 60°
3. 30°
4. 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°.