Triangles - ACT Math

In triangle MNO, angle M measures 120° and angle N measures 30°. What is the measure of angle O?
Answer: 30° We need to find angle O in triangle MNO where angle M = 120° and angle N = 30°. Using the triangle angle sum theorem, all angles sum to 180°. So angle O = 180° - 120° - 30° = 30°. This triangle has two 30° angles, making it isosceles.
In a $30$ - $60$ - $90$ right triangle, the hypotenuse has length $18$ . What is the length of the shortest side (opposite $30^\circ$ )?
Answer: $9$ In a 30-60-90 triangle, we need to find the shortest side when the hypotenuse is 18. In a 30-60-90 triangle, the sides are in the ratio 1:√3:2, where the shortest side is half the hypotenuse. Therefore, shortest side = 18 ÷ 2 = 9. Choice B gives 6√3 ≈ 10.39, which would be the longer leg.
If two angles of a triangle measure $38^\circ$ and $90^\circ$ , what is the measure of the third angle?
Answer: $52^\circ$ 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.
In a right triangle, if one leg measures 9 units and the other leg measures 12 units, what is the length of the hypotenuse?
Answer: 15 units We need to find the hypotenuse of a right triangle with legs measuring 9 and 12 units. Using the Pythagorean theorem: a² + b² = c². So 9² + 12² = c², which gives 81 + 144 = 225, therefore c = √225 = 15 units. This is the classic 3-4-5 triangle scaled up by 3.
What type of triangle has all sides of different lengths?
Answer: Scalene We need to identify the type of triangle with all sides of different lengths. A scalene triangle is defined as having all three sides of different lengths, which also means all three angles are different. Equilateral triangles have all sides equal, while isosceles triangles have two equal sides.
In triangle VWX, angle V measures 30°, and angle W measures 60°. What is the measure of angle X?
Answer: 90° 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°.
In a 45-45-90 triangle, if one leg measures 7 units, what is the length of the hypotenuse?
Answer: 7√2 units We need to find the hypotenuse of a 45-45-90 triangle with one leg measuring 7 units. In a 45-45-90 triangle, both legs are equal and the hypotenuse = leg × √2. With a leg of 7 units, the hypotenuse = 7√2 units. Choice C (14 units) incorrectly doubles the leg instead of multiplying by √2.
Triangle $XYZ$ has side lengths $10$ , $10$ , and $10$ . Which describes the triangle? (Choose from: right, isosceles, equilateral, scalene.)
Answer: Equilateral We need to classify triangle XYZ with side lengths 10, 10, and 10. Since all three sides have the same length, this triangle has all sides equal. A triangle with all three sides equal is called an equilateral triangle. An equilateral triangle also has all angles equal to 60° each.
Triangle $DEF$ has angles $65^\circ$ , $65^\circ$ , and $x$ . What is the measure of angle $x$ ?
Answer: $50^\circ$ We need to find angle x in triangle DEF where two angles measure 65° each. The sum of angles in any triangle is 180°. Setting up the equation: 65° + 65° + x = 180°, so 130° + x = 180°, which gives x = 50°. Since two angles are equal (65° each), this confirms the triangle is isosceles.
In a right triangle, one leg measures 8 units and the hypotenuse is 10 units. What is the length of the other leg?
Answer: 6 units 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.
In right triangle PQR, angle P is the right angle. If side PQ = 5 and side PR = 12, what is the length of side QR?
Answer: 13 We need to find the length of hypotenuse QR in right triangle PQR. Since P is the right angle, sides PQ and PR are the legs, and QR is the hypotenuse. Using the Pythagorean theorem: PQ² + PR² = QR², so 5² + 12² = QR². Calculating: 25 + 144 = 169, therefore QR = √169 = 13.
In triangle PQR, angle P measures 150° and angle Q measures 15°. What is the measure of angle R?
Answer: 15° We need to find angle R in triangle PQR where angle P = 150° and angle Q = 15°. Using the triangle angle sum theorem, all angles sum to 180°. So angle R = 180° - 150° - 15° = 15°. This triangle is obtuse (angle P > 90°) and isosceles (angles Q and R are equal).
A right triangle has legs of lengths $9$ and $12$ . What is the hypotenuse of the triangle? Give your answer in simplest form.
Answer: $15$ We need to find the hypotenuse of a right triangle with legs 9 and 12. Using the Pythagorean theorem: $9^2 + 12^2 = c^2$ . Calculating: $81 + 144 = 225$ , so $c^2 = 225$ and $c = 15$ . Choice B shows $\sqrt{225}$ , which equals 15 but is not simplified, and choice D shows $\sqrt{369}$ which would be incorrect calculation.
In triangle GHI, angle G measures 45° and angle H measures 45°. What type of triangle is GHI?
Answer: Isosceles 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°.
Triangle STU is a 45-45-90 triangle. If the hypotenuse ST is 10, what is the length of each leg?
Answer: $5\sqrt{2}$ 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 $\sqrt{2}$ .
In triangle GHI, angle G measures 100° and angle H measures 40°. What is the measure of angle I?
Answer: 40° We need to find angle I in triangle GHI where angle G = 100° and angle H = 40°. Using the triangle angle sum theorem, all angles sum to 180°. So angle I = 180° - 100° - 40° = 40°. This triangle is obtuse (angle G > 90°) and isosceles (angles H and I are equal).
A triangle has side lengths $7$ , $24$ , and $25$ . Which describes the triangle? (Choose from: right, isosceles, equilateral, scalene.)
Answer: Right We need to classify a triangle with side lengths 7, 24, and 25. To check if it's a right triangle, we test if a² + b² = c² using the largest side as the potential hypotenuse. Calculating: 7² + 24² = 49 + 576 = 625, and 25² = 625. Since the equation holds true, this is a right triangle.
A triangle has angles $44^\circ$ and $89^\circ$ . What is the measure of the third angle?
Answer: $47^\circ$ We need to find the third angle when two angles measure 44° and 89°. The sum of angles in any triangle is 180°. Setting up the equation: 44° + 89° + x = 180°, so 133° + x = 180°, which gives x = 47°. Choice B shows 137°, but this would make the total exceed 180°.
Triangle ABC is a 30-60-90 triangle. If side AC (the hypotenuse) is 10, what is the length of side AB (shorter leg)?
Answer: 5 We need to find the shorter leg AB in a 30-60-90 triangle with hypotenuse AC = 10. In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2, where the shortest side is opposite the 30° angle. Since the hypotenuse is 10, the shorter leg equals 10 ÷ 2 = 5. The longer leg would be 5√3.
In triangle STU, if ST = TU = US, what type of triangle is triangle STU?
Answer: Equilateral We need to identify the type of triangle STU where ST = TU = US. When all three sides of a triangle are equal in length, it is called an equilateral triangle. This also means all three angles are equal, each measuring 60°. An isosceles triangle has only two equal sides.

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