Angle Relationships - SSAT Upper Level Quantitative

1. Triangle $ABC$ has an exterior angle at vertex $B$ that measures $142°$. If angle $A$ is $28°$ more than angle $C$, what is the measure of angle $A$?
   Answer: 85° An exterior angle of a triangle equals the sum of the two non-adjacent interior angles. So the exterior angle at B equals angles A and C: A + C = 142°. Given that A = C + 28°, substitute: (C + 28°) + C = 142°, so 2C + 28° = 142°, giving 2C = 114° and C = 57°. Therefore A = 57° + 28° = 85°. Choice A (57°) is the value of angle C. Choice C (95°) results from incorrectly using A = C - 28°. Choice D (104°) comes from using the exterior angle as an interior angle.
2. In quadrilateral $PQRS$, the sum of three angles is $285°$. If angle $P$ is twice angle $Q$, and angle $R$ equals angle $Q$ plus $15°$, what is the measure of angle $S$?
   Answer: 75° In any quadrilateral, the sum of interior angles is 360°. We're told three angles sum to 285°, so the fourth angle S = 360° - 285° = 75°. The other relationships given (P = 2Q, R = Q + 15°) are extra information that might be used to verify, but aren't needed to find S directly. Choice A (60°) might result from incorrectly using 345° as the sum. Choice C (90°) might come from using 270° as the total. Choice D (105°) could result from calculation errors.
3. Quadrilateral $MNPQ$ is inscribed in a circle. If angle $M = 3y + 15°$, angle $N = 2y + 25°$, and angle $P = 4y - 10°$, what is the measure of angle $Q$?
   Answer: 105° In a cyclic quadrilateral (inscribed in a circle), opposite angles are supplementary. So angle M + angle P = 180° and angle N + angle Q = 180°. From the first relationship: (3y + 15°) + (4y - 10°) = 180°, so 7y + 5° = 180°, giving 7y = 175° and y = 25°. Now angle N = 2(25°) + 25° = 50° + 25° = 75°. Since angle N + angle Q = 180°, we have angle Q = 180° - 75° = 105°. We can verify: M = 3(25°) + 15° = 90° and P = 4(25°) - 10° = 90°, so M + P = 180° ✓.
4. Two parallel lines are cut by two different transversals. The first transversal creates an angle of $58°$ with the first parallel line. The second transversal creates corresponding angles, one of which is $3x + 14°$ and another which is $5x - 22°$. What is the value of $x$?
   Answer: 18 When parallel lines are cut by a transversal, corresponding angles are equal. Therefore, the two corresponding angles created by the second transversal must be equal: 3x + 14° = 5x - 22°. Solving: 14° + 22° = 5x - 3x, so 36° = 2x, giving x = 18. The information about the first transversal creating a 58° angle is extra information not needed for this problem. Choice A (15) results from setting up 3x + 14° + 5x - 22° = 180° incorrectly. Choice B (16) comes from arithmetic errors. Choice D (20) might result from using the 58° angle incorrectly in calculations.

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