Angle questions on Digital SAT Math draw from a handful of core facts: a polygon's interior angles sum to (n − 2) × 180°, corresponding angles formed by a transversal crossing parallel lines are equal, an exterior angle of a triangle equals the sum of the two remote interior angles, and complementary acute angles satisfy cos(A) = sin(90° − A). Similar figures preserve angle measure regardless of any difference in side length or scale factor — a detail that trips up students who assume similarity affects angles as well as sides.
The 14 problems below cover this full range, including polygon angle sums, parallel-line transversal relationships, triangle exterior angle problems, similar-figure angle correspondence, and radian-degree conversions. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. When two figures are described as similar, remember that corresponding angles are always exactly equal — the given side lengths in those problems are usually there to test whether you get distracted by them rather than to actually change your answer. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
1Finding a Polygon's Interior Angle Constant
A polygon has exactly 73 sides. If the measure of each of the 73 interior angles of this polygon is (180p)°, what is the value of p?
Answer Explanation
The sum of the interior angles of an n-sided polygon is (n − 2) × 180°. With 73 sides, that sum is (73 − 2) × 180 = 71 × 180.
Since each of the 73 equal angles measures (180p)°, the total sum is also 73 × 180p. Setting these equal: 73(180p) = 71(180). Dividing both sides by 180: 73p = 71, so p = 71/73.
2Using Complementary Acute Angles
For two acute angles, ∠A and ∠B, cos A = sin B. The measures, in degrees, of ∠A and ∠B are x + 8 and 4x + 2, respectively. What is the value of x?
A) 37
B) 34
C) 16
D) 2
Answer Explanation
For acute angles, cos A = sin B exactly when A and B are complementary (A + B = 90°), since cos A = sin(90° − A).
Set up the equation: (x + 8) + (4x + 2) = 90. Combine like terms: 5x + 10 = 90, so 5x = 80, giving x = 16. The correct answer is C.
3Finding an Angle in Similar Quadrilaterals
Quadrilaterals PQRS and WXYZ are similar, where P, Q, and R correspond to W, X, and Y, respectively. The measure of ∠S is 105°, PS = 25, and WZ = 5. What is the measure of ∠Z?
A) 5°
B) 21°
C) 75°
D) 105°
Answer Explanation
Since P↔W, Q↔X, and R↔Y are given as corresponding vertices, the remaining vertex correspondence must be S↔Z. Corresponding angles in similar figures are always exactly equal, regardless of the scale factor between the side lengths.
So ∠Z = ∠S = 105°. The given side lengths (PS = 25, WZ = 5) are not needed to answer this question. The correct answer is D.
4Using Complementary Acute Angles
For two acute angles, ∠Q and ∠R, cos Q = sin R. The measures, in degrees, of ∠Q and ∠R are x + 21 and 4x + 9, respectively. What is the value of x?
A) 4
B) 12
C) 30
D) 69
Answer Explanation
Since cos Q = sin R for acute angles exactly when Q and R are complementary, set up: (x + 21) + (4x + 9) = 90.
Combine like terms: 5x + 30 = 90, so 5x = 60, giving x = 12. The correct answer is B.
5Identifying an Impossible Angle Sum
Two lines intersect at exactly one point, forming two acute angles and two obtuse angles. The measure of one of these angles is (9x − 190)°. Which of the following could NOT be the sum of the measures of any two of these angles?
A) (−18x + 380)°
B) (−18x + 740)°
C) (18x − 380)°
D) 180°
Answer Explanation
Two intersecting lines create only two distinct angle measures (each repeated via vertical angles): the given angle (9x−190) and its supplement, 180−(9x−190) = 370−9x. Any two-angle sum is either: two copies of the first angle (18x−380), two copies of the second (740−18x), or one of each, which always totals exactly 180° regardless of x.
Option B (−18x+740) matches 740−18x exactly, option C (18x−380) matches the first sum exactly, and option D (180°) matches the mixed-pair sum exactly. Option A (−18x+380) is the negation of option C's expression and doesn't match any valid sum pattern. The correct answer is A.
6Converting Radians to Degrees
The measure of angle R is 2π/3 radians. The measure of angle T is 13π/18 radians greater than the measure of angle R. What is the measure of angle T, in degrees?
A) 120
B) 130
C) 250
D) 500
Answer Explanation
Find angle T in radians first: T = 2π/3 + 13π/18. Convert 2π/3 to eighteenths: 2π/3 = 12π/18. So T = 12π/18 + 13π/18 = 25π/18 radians.
Convert to degrees by multiplying by 180/π: (25π/18)(180/π) = 25 × 10 = 250°. The correct answer is C.
7Finding an Angle in Similar Quadrilaterals
Quadrilaterals PQRS and WXYZ are similar where P, Q, and R correspond to W, X, and Y, respectively. The measure of ∠S is 135°, PS = 45, and WZ = 9. What is the measure of ∠Z?
A) 5°
B) 27°
C) 45°
D) 135°
Answer Explanation
Since P↔W, Q↔X, and R↔Y, the remaining correspondence is S↔Z. Corresponding angles in similar figures are equal, independent of the side-length scale factor.
So ∠Z = ∠S = 135°. The correct answer is D.
8Solving a Parallel-Line Angle Relationship
In a figure, line r is parallel to line s, and both lines are intersected by line t. If x = 2w + 24, y = 5w + 29, and 7w = a, where a is a constant, and x and y are the same-side interior angles formed at the two intersections, what is the value of a?
Note: figure not drawn to scale; x and y are same-side interior angles
Answer Explanation
Same-side interior angles formed by a transversal crossing two parallel lines are supplementary (they sum to 180°): (2w + 24) + (5w + 29) = 180.
Combine like terms: 7w + 53 = 180, so 7w = 127. Since a = 7w directly, a = 127.
9Finding a Polygon's Interior Angle Constant
A polygon has exactly 83 sides. If the measures of each of the 83 interior angles of this polygon is (180p)°, what value of p?
Answer Explanation
The sum of the interior angles of an 83-sided polygon is (83 − 2) × 180 = 81 × 180.
Since 83 equal angles of (180p)° each must total this sum: 83(180p) = 81(180). Dividing both sides by 180: 83p = 81, so p = 81/83.
10Finding Vertical Angles
In the figure, two lines intersect at a point. If w = 150, what is the value of z?
A) 30
B) 50
C) 75
D) 150
Answer Explanation
When two lines cross at a point, they form two pairs of vertical angles — angles directly across from each other through the intersection point, which are always equal.
Since w and z are positioned as vertical angles in this figure, z = w = 150. The correct answer is D.
11Finding an Angle in Similar Quadrilaterals with a Scale Factor
Quadrilateral P'Q'R'S' is similar to quadrilateral PQRS where P, Q, R, and S correspond to P', Q', R', and S', respectively. The measure of angle P is 45°, the measure of angle Q is 70°, and the measure of angle R is 90°. The length of each side of P'Q'R'S' is 3 times the length of each corresponding side of PQRS. What is the measure of angle P'?
A) 15°
B) 45°
C) 60°
D) 135°
Answer Explanation
Corresponding angles in similar figures are always equal, no matter what the scale factor between the side lengths is (the 3× side-length scaling here doesn't affect angle measure at all).
Since P corresponds to P', ∠P' = ∠P = 45°. The correct answer is B.
12Converting a Radian Difference to Degrees
The difference between the measure of angle A and the measure of angle B is −3/4 π radians. Which expression shows the difference between the measure of angle A and the measure of angle B, in degrees?
A) −3/4 · 180°/π
B) −3/4π · π/180°
C) −3/4π · 180°/π
D) −3/4π · 360°/π
Answer Explanation
To convert a radian measure to degrees, multiply by the conversion factor 180°/π. The given radian difference is −3/4π.
Applying the conversion factor to the full radian value: (−3/4π) × (180°/π). The correct answer is C.
13Solving a Parallel-Line Angle Relationship
In a figure, line n intersects lines r and s, where line r is parallel to line s. One angle formed at the intersection with r measures 138°, and the corresponding angle x° is formed at the intersection with s. What is the value of x?
Note: figure not drawn to scale
Answer Explanation
When a transversal crosses two parallel lines, corresponding angles are equal.
So x = 138.
14Using the Exterior Angle Theorem with a System of Equations
In triangle PQR, the measure of angle P is (3x+5)°, the measure of angle Q is (2x+9)°, and the measure of angle R is (4y+5)°. If side QR is extended through point R to point S, and the measure of angle PRS is (x+y)°, what is the value of x + y?
Note: figure not drawn to scale
Answer Explanation
By the exterior angle theorem, the exterior angle PRS equals the sum of the two remote interior angles: x + y = (3x+5) + (2x+9) = 5x + 14.
Also, angle PRS and the interior angle at R (4y+5) are supplementary (they form a straight line along the extended side): (x+y) + (4y+5) = 180, giving x + 5y = 175. Combined with the triangle's angle sum, (3x+5)+(2x+9)+(4y+5)=180, giving 5x+4y=161. Solving this system: from x+y=5x+14, y=4x+14; substituting into x+5y=175 gives x+5(4x+14)=175, so 21x+70=175, giving x=5. Then y=4(5)+14=34. Check: 5(5)+4(34)=25+136=161 ✓. So x+y = 5+34 = 39.
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