This question tests reasoning in an equilateral triangle with a perpendicular. The governing geometric principle is that in an equilateral triangle, all angles are 60°, and the altitude to the base forms right angles at the base. With AD perpendicular to BC, angle ADB is the angle at D, which is 90°. This follows from the definition of perpendicular lines. The result is justified as AD is explicitly perpendicular to BC. A representative distractor like 60° might come from confusing the right angle with a vertex angle of the equilateral triangle.

This question tests the properties of parallel lines cut by a transversal. When parallel lines are cut by a transversal, consecutive interior angles (also called co-interior or same-side interior angles) are supplementary, meaning they sum to 180°. Given that one interior angle on the same side of the transversal measures 104°, the other interior angle on the same side must measure 180° - 104° = 76°. Choice A (104°) incorrectly assumes these angles are equal, which would only be true for alternate interior angles. Choice D (52°) might result from halving the given angle rather than finding its supplement.

This question tests the concept of vertical angles formed by intersecting lines. When two lines intersect, they form two pairs of vertical angles, which are the angles opposite each other. The Vertical Angles Theorem states that vertical angles are always congruent (equal in measure). Therefore, if one angle measures 125°, its vertical angle also measures 125°. Choice A (55°) represents the supplement of 125°, which would be an adjacent angle, not the vertical angle. Choice D (25°) has no geometric relationship to the given angle.

This question tests understanding of angles formed by intersecting lines. When two lines intersect, they form two pairs of vertical angles and adjacent angles that are supplementary. Adjacent angles share a common side and together form a straight line, so they sum to 180°. Since one angle measures 38°, its adjacent angle must measure 180° - 38° = 142°. Choice A (38°) incorrectly assumes adjacent angles are equal, when only vertical angles are equal. Choice B (52°) might result from misunderstanding the relationship between adjacent angles.

This question tests right triangle reasoning. The governing geometric principle is the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. The legs are AC = 6 and BC = 8, so AB = √(6² + 8²) = √(36 + 64) = √100 = 10. This calculation directly applies the theorem to find the hypotenuse. The result is justified as it satisfies the right triangle property. A representative distractor like √52 might result from mistakenly using half of one leg or assuming incorrect lengths.

This question tests exterior angle reasoning in a triangle. The governing geometric principle is that an exterior angle of a triangle equals the sum of the two remote interior angles. The exterior angle at C is 132°, and angle A is 47°, so angle B = 132° - 47° = 85°. This calculation applies the exterior angle theorem directly. The result is justified as it matches the sum of the remote interiors. A representative distractor like 95° might arise from subtracting from 180° instead of using the exterior theorem.

This question tests reasoning in an isosceles triangle with a perpendicular bisector. The governing geometric principle is that in an isosceles triangle, the altitude from the vertex to the base bisects the vertex angle. With AB = AC and angle BAC = 40°, the altitude AD bisects it into two 20° angles. Thus, angle BAD = 20°. This result is justified because the altitude coincides with the angle bisector in an isosceles triangle. A representative distractor like 30° might result from incorrectly assuming the triangle is equilateral or miscalculating base angles.

This question tests triangle angle sum and angle bisector reasoning. The governing geometric principle is that the sum of angles in a triangle is 180 degrees, and an angle bisector divides the vertex angle into two equal parts. In triangle ABC, the angles at B and C are 50° and 70°, so the angle at A is 180° - 50° - 70° = 60°. Since AD bisects angle BAC, angle BAD equals half of 60°, which is 30°. This result is justified because the bisector creates two congruent angles from the vertex angle. A representative distractor like 35° might arise from miscalculating the angle at A as 65° by incorrectly subtracting the given angles.

This question tests reasoning in an isosceles triangle with a perpendicular altitude. The governing geometric principle is the Pythagorean theorem applied to the right triangles formed by the altitude to the base. With AB = AC = 10 and BC = 12, the altitude AD splits BC into two segments of 6 each, so AD = √(10² - 6²) = √(100 - 36) = √64 = 8. This calculation uses the theorem in half the triangle. The result is justified as it fits the geometry of the isosceles triangle. A representative distractor like 6 might come from assuming AD equals half the base instead of calculating the height.

This question tests angle reasoning at the intersection of lines. The governing geometric principle is that adjacent angles at an intersection form a linear pair and sum to 180 degrees. One angle is 92°, so the adjacent angle is 180° - 92° = 88°. This applies the linear pair property directly. The result is justified as adjacent angles on a straight line are supplementary. A representative distractor like 92° might result from confusing adjacent with vertical angles, which are equal.