Triangle Angle Sum - SHSAT Math

1. In the figure shown, $\triangle ABC$ has $\angle ABC = 90°$. A point $D$ on segment $AC$ satisfies $BD \perp AC$. If $\angle BAC = (2x+15)°$ and $\angle DBC = (3x-5)°$, what is the measure of $\angle BCA$?
   Answer: $35°$ In right $\triangle ABD$: $\angle ABD = 90°-\angle BAD = 90°-(2x+15)=75-2x$. Since $\angle ABD+\angle DBC=90°$: $(75-2x)+(3x-5)=90 \Rightarrow x+70=90 \Rightarrow x=20$. So $\angle BAC=55°$, $\angle BCA=180°-90°-55°=35°$. Distractors: (B) 40°=$3x-5$ with wrong x; (C) 50° mix-up; (D) 55°=angle A.
2. In $\triangle PQR,$ the measures of $\angle P$ and $\angle Q$ are $50^{\circ}$ and $60^{\circ},$ respectively. What is the measure of $\angle R$?
   Answer: $70^{\circ}$ When you encounter a triangle angle problem, remember that the sum of all three interior angles in any triangle is always $180°$. This is one of the most fundamental rules in geometry. Given that $\angle P = 50°$ and $\angle Q = 60°$, you can find $\angle R$ by setting up the equation: $\angle P + \angle Q + \angle R = 180°$ Substituting the known values: $50° + 60° + \angle R = 180°$ Simplifying: $110° + \angle R = 180°$ Solving for $\angle R$: $\angle R = 180° - 110° = 70°$ Looking at the wrong answers: Choice B ($80°$) would make the triangle's angles sum to $190°$, which is impossible. Choice C ($110°$) is actually the sum of the two given angles—a common mistake where students add instead of subtract. Choice D ($120°$) would create a triangle with angles totaling $230°$, again violating the triangle angle sum theorem. The correct answer is A ($70°$). Strategy tip: Always check your work by adding all three angles together—they must equal exactly $180°$. If you get any other sum, you've made an error. Also, watch out for the trap of confusing the sum of known angles with the missing angle itself.
3. Triangle $ABC$ has angles in the ratio $7:9:14$. If triangle $DEF$ has angles in the same ratio and the smallest angle of triangle $DEF$ measures $42°$, what is the measure of the largest angle in triangle $DEF$?
   Answer: $84°$ The angles of triangle $DEF$ are $7k$, $9k$ and $14k$ for some number $k$. The smallest is $7k = 42°$, so $k = 6°$. The angles are $7(6) = 42°$, $9(6) = 54°$ and $14(6) = 84°$. Check: $42° + 54° + 84° = 180°$, as every triangle requires. The largest angle measures $84°$. The value $96°$ is the sum of the two smaller angles, $42° + 54°$, not the largest angle itself. The value $80°$ comes from assuming the angles must be $40°$, $60°$ and $80°$, ignoring the given ratio and the $42°$ angle. The value $88°$ does not fit the ratio: it is not $14k$ for $k = 6°$, and with $42°$ and $54°$ it would make the angles total $184°$.
4. In triangle $ABC$, the measure of angle $A$ is $3x + 10°$, the measure of angle $B$ is $2x - 5°$, and the measure of angle $C$ is $x + 25°$. What is the value of $x$?
   Answer: $25°$ Using the triangle angle sum theorem, the sum of all angles equals 180°. Setting up the equation: $(3x + 10) + (2x - 5) + (x + 25) = 180$. Simplifying: $6x + 30 = 180$, so $6x = 150$ and $x = 25°$. Choice B results from incorrectly setting the sum to 175°. Choice C comes from an algebraic error when combining like terms. Choice D results from setting the sum to 210° instead of 180°.
5. In triangle $RST$, the measure of an exterior angle at vertex $S$ is $140°$. If angle $R$ is $25°$ more than angle $T$, what is the measure of angle $T$?
   Answer: $57.5°$ The exterior angle at $S$ equals the sum of the two non-adjacent interior angles: angle $R$ + angle $T = 140°$. Since angle $R =$ angle $T + 25°$, we can substitute: $(T + 25) + T = 140°$. This gives $2T + 25 = 140°$, so $2T = 115°$ and $T = 57.5°$. Choice B results from incorrectly setting up the equation as $T + 25 = 140°$. Choice C comes from assuming the angles are complementary rather than using the exterior angle theorem. Choice D results from calculation errors.
6. Triangle $XYZ$ is isosceles with $XY = XZ$. If the vertex angle at $X$ measures $38°$, and triangle $XYZ$ is divided into two smaller triangles by drawing altitude $XM$ from $X$ to side $YZ$, what is the measure of angle $YXM$?
   Answer: $19°$ In an isosceles triangle, the altitude from the vertex angle to the base also bisects the vertex angle. Since the vertex angle $X = 38°$, the altitude $XM$ divides this angle into two equal parts: angle $YXM = 38° ÷ 2 = 19°$. Choice B gives the measure of base angle $XYZ$ (since base angles = $(180° - 38°) ÷ 2 = 71°$). Choice C incorrectly gives the full vertex angle. Choice D is incorrect and doesn't correspond to any angle in the triangle.

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