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.

We need to find the measure of angle C in triangle ABC. Using the triangle angle sum theorem, the sum of all three angles in any triangle equals 180°. Setting up the equation: 90° + 45° + C = 180°, so 135° + C = 180°, which gives us C = 45°.

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.

We need to find the hypotenuse of a right triangle with legs measuring 3 and 4 units. Using the Pythagorean theorem, a² + b² = c² where c is the hypotenuse. Substituting: 3² + 4² = c², so 9 + 16 = 25, therefore c = √25 = 5. This is the classic 3-4-5 right triangle.

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.

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).

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.

We need to find the leg length of a 45-45-90 triangle with $hypotenuse = 10$ units. In a 45-45-90 triangle, $hypotenuse = leg \times \sqrt{2}$, so $leg = \frac{hypotenuse}{\sqrt{2}}$. With hypotenuse = 10, each leg = $10 \div \sqrt{2} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}$ units. Choice D (10 units) incorrectly uses the hypotenuse value.

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.

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.