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).

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.

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°.

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.

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.

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.

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.

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.

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$\sqrt{2}$.

We need to classify triangle MNO with sides 5, 12, and 13. To determine if it's a right triangle, we check if the Pythagorean theorem holds: 5² + 12² = 13². Calculating: 25 + 144 = 169, and 13² = 169. Since the equation holds true (25 + 144 = 169), this is a right triangle with the right angle opposite the longest side (13).