This question requires finding the measure of angle C in triangle ABC, where angles A and B are 42 degrees and 71 degrees, respectively. The key property is that the sum of the interior angles in any triangle is 180 degrees. To solve, add angles A and B: 42 + 71 = 113, then subtract from 180 to get angle C = 67 degrees. This direct calculation uses the triangle angle sum theorem. A common error is forgetting to subtract from 180, perhaps adding all three incorrectly or miscalculating the arithmetic. Another mistake might involve confusing this with exterior angles. A useful strategy is to quickly sum the given angles and subtract from 180 to verify the third angle.

This question involves vertical angles, which are always congruent. Since the angles (5x - 15)° and (3x + 25)° are vertical angles, they must be equal. Setting up the equation: 5x - 15 = 3x + 25. Solving: 5x - 3x = 25 + 15, which gives 2x = 40, so x = 20. To verify: when x = 20, the first angle is 5(20) - 15 = 85°, and the second is 3(20) + 25 = 85°, confirming they're equal. A common error is thinking vertical angles are supplementary rather than congruent. Remember that vertical angles are formed by intersecting lines and are always equal.

This question involves an isosceles triangle with congruent sides JK and JL. In an isosceles triangle, the base angles (opposite the congruent sides) are congruent. The vertex angle KJL = 40°, and we need to find base angle JKL. Since the sum of angles in a triangle is 180°, and the two base angles are equal, we have: 40° + 2(base angle) = 180°. This gives us 2(base angle) = 140°, so each base angle = 70°. A common mistake is confusing which angles are the base angles in an isosceles triangle. Remember that base angles are opposite the congruent sides and are always equal to each other.

This question asks for an angle measure in a triangle given coordinates. Points A(-2,1), B(4,1), and C(4,5) form a triangle. Notice that A and B have the same y-coordinate (1), so AB is horizontal. Also, B and C have the same x-coordinate (4), so BC is vertical. When a horizontal line meets a vertical line, they form a 90° angle. Therefore, angle B = 90°. We can verify this is a right triangle by checking that AB is perpendicular to BC. A common mistake is trying to use distance formula when the angle can be determined by recognizing perpendicular lines. On coordinate geometry problems, always check if sides are horizontal or vertical first.

This question involves an isosceles triangle where PQ = PR. In an isosceles triangle, the base angles (angles opposite the congruent sides) are congruent. Since P is the vertex angle measuring 40°, angles Q and R are the base angles. Using the triangle angle sum: 40° + angle Q + angle R = 180°. Since angle Q = angle R in an isosceles triangle, we have 40° + 2(angle Q) = 180°. Solving: 2(angle Q) = 140°, so angle Q = 70°. A common mistake is thinking the vertex angle should be split equally between the base angles. Remember that in isosceles triangles, it's the base angles that are equal, not halves of the vertex angle.

This question asks for angle B in right triangle ABC where angle C = 90° and angle A = 28°. In any triangle, the sum of all interior angles equals 180°. For this right triangle: angle A + angle B + angle C = 180°. Substituting: 28° + angle B + 90° = 180°. Solving: angle B = 180° - 118° = 62°. Therefore, angle B = 62°. A common mistake would be forgetting to include all three angles in the sum or making arithmetic errors when subtracting.

This question asks which value of x makes triangle PQR valid using the triangle inequality theorem. The triangle inequality states that the sum of any two sides must be greater than the third side. For sides PQ = 7, QR = 10, and PR = x, we need: 7 + 10 > x, 7 + x > 10, and 10 + x > 7. This gives us: x < 17, x > 3, and x > -3. Since x must be positive, we need 3 < x < 17. Among the choices, x = 12 satisfies this condition (3 < 12 < 17). The other values either violate the triangle inequality (x = 2, 3, 17) or don't form a valid triangle.

This question asks which triangle inequality must be true for triangle ABC with sides AB = 9, BC = 11, and AC = 15. The triangle inequality theorem states that the sum of any two sides must be greater than the third side. We need to check all three inequalities: 9 + 11 > 15 gives 20 > 15 ✓, 9 + 15 > 11 gives 24 > 11 ✓, and 11 + 15 > 9 gives 26 > 9 ✓. Among the choices, only B (9 + 15 > 11) represents a correct triangle inequality. The other choices either show false inequalities or incorrect relationships.

This question asks for the measure of angle R in equilateral triangle RST. An equilateral triangle has all three sides equal and all three angles equal. Since the sum of angles in any triangle is 180°, each angle in an equilateral triangle measures 180° ÷ 3 = 60°. Therefore, angle R = 60°. A common mistake would be confusing equilateral triangles with isosceles triangles or forgetting that all angles in an equilateral triangle are 60°. This is a fundamental property that should be memorized.

This question asks for the length of segment QR using the distance formula. Given points P(0,0), Q(6,0), and R(0,8), we need to find the distance from Q(6,0) to R(0,8). Using the distance formula: QR = √[(0-6)² + (8-0)²] = √[(-6)² + 8²] = √[36 + 64] = √100 = 10. Therefore, the length of QR is 10. A common mistake would be incorrectly substituting coordinates into the distance formula or making arithmetic errors when calculating squares and square roots.