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 asks about vertical angles formed when two lines intersect. Vertical angles are the pairs of opposite angles formed when two lines intersect, and they are always congruent. Since one angle measures 47°, the angle vertical to it (directly opposite across the intersection point) also measures 47°. The other two angles at the intersection would each measure 180° - 47° = 133°, as adjacent angles are supplementary. A common error is confusing vertical angles with adjacent angles or thinking vertical angles are supplementary. Remember: vertical angles are always equal, while adjacent angles on a straight line are supplementary.

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 the length of vertical segment AB on coordinate plane. Given points A(2,3) and B(2,-5), both points have the same x-coordinate (x = 2), making this a vertical segment. For vertical segments, the distance is the absolute value of the difference in y-coordinates: |3 - (-5)| = |8| = 8. Therefore, the length of segment AB is 8. A common mistake would be unnecessarily using the full distance formula when the segment is clearly vertical or horizontal.

This question asks for the length of B'C' when triangles ABC and A'B'C' are similar with scale factor 3/2. When triangles are similar, corresponding sides are proportional to the scale factor. Given BC = 8 in triangle ABC and the scale factor from ABC to A'B'C' is 3/2, the corresponding side B'C' equals: B'C' = BC × (scale factor) = 8 × (3/2) = 12. Therefore, B'C' = 12. A common mistake would be using the reciprocal of the scale factor or confusing which triangle is being scaled to which.

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.