SHSAT Math Quiz: Triangle Angle Sum

6 questions · exam conditions

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Triangle Angle SumQuestion 1 of 6

In $\triangle PQR,$ the measures of $\angle P$ and $\angle Q$ are $50^{\circ}$ and $60^{\circ},$ respectively. What is the measure of $\angle R$ ?

70^{\circ}

80^{\circ}

110^{\circ}

120^{\circ}

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SHSAT Math Quiz

SHSAT Math Quiz: Triangle Angle Sum

Practice Triangle Angle Sum in SHSAT Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Triangle Angle Sum, giving you a quick way to practice the rules, question types, and explanations that matter most for SHSAT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

In $\triangle PQR,$ the measures of $\angle P$ and $\angle Q$ are $50^{\circ}$ and $60^{\circ},$ respectively. What is the measure of $\angle R$ ?

$70^{\circ}$ (correct answer)
$80^{\circ}$
$110^{\circ}$
$120^{\circ}$

Explanation: 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.

Question 2

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 $YXM$ ?

$19°$ (correct answer)
$71°$
$38°$
$52°$

Explanation: 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.

Question 3

Triangle $ABC$ has angles in the ratio $2:3:4$ . 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 $DEF$ ?

$84°$ (correct answer)
$96°$
$80°$
$88°$

Explanation: If triangle

DEF

has angles in the ratio

2:3:4

and the smallest angle is

42°

, then the smallest angle corresponds to the

2x

term in the ratio. Setting

2x = 42°

gives

x = 21°

. Therefore, the three angles are

2x = 42°

3x = 63°

, and

4x = 84°

. The largest angle measures

84°

. We can verify:

42° + 63° + 84° = 189°

. Wait, this should be

180°

. Let me recalculate: if the angles are in ratio

2:3:4

, then

2k + 3k + 4k = 180°

where

k = 20°

. So the angles are

40°, 60°, 80°

. If the smallest angle in triangle

DEF

42°

, then scaling factor is

42°/40° = 1.05

. The largest angle is

80° × 1.05 = 84°

Question 4

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 $\angle BCA$ ?

$35°$ (correct answer)
$40°$
$50°$
$55°$

Explanation: 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.

Question 5

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 $x$ ?

$25°$ (correct answer)
$30°$
$20°$
$35°$

Explanation: 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°.

Question 6

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 $T$ ?

$57.5°$ (correct answer)
$65°$
$40°$
$75°$

Explanation: 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.