ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Angle Relationships And Polygons
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Angle Relationships And PolygonsQuestion 1 of 19

The measure of an angle's supplement is 10 degrees less than three times the measure of its complement. What is the measure of the angle?

35°
40°
50°
80°
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ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Angle Relationships And Polygons

Practice Angle Relationships And Polygons in ACCUPLACER Quantitative Reasoning, Algebra & Statistics 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 Angle Relationships And Polygons, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Quantitative Reasoning, Algebra & Statistics.

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.

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Question 1

The measure of an angle's supplement is 10 degrees less than three times the measure of its complement. What is the measure of the angle?

  1. 35°
  2. 40° (correct answer)
  3. 50°
  4. 80°
Explanation: Let the angle be xx. Its supplement is 180−x180 - x and its complement is 90−x90 - x. The problem states that the supplement is 10 less than three times the complement. This translates to the equation: 180−x=3(90−x)−10180 - x = 3(90 - x) - 10. Distribute the 3: 180−x=270−3x−10180 - x = 270 - 3x - 10. Simplify: 180−x=260−3x180 - x = 260 - 3x. Add 3x3x to both sides: 180+2x=260180 + 2x = 260. Subtract 180 from both sides: 2x=802x = 80. Solve for xx: x=40°x = 40°.

Question 2

Four of the interior angles of a pentagon measure 90°, 100°, 105°, and 115°. What is the measure of the fifth angle?

  1. 130° (correct answer)
  2. 120°
  3. 135°
  4. 140°
Explanation: First, determine the sum of the interior angles of a pentagon (a polygon with 5 sides) using the formula (n−2)×180°(n-2) \times 180°. For a pentagon, this is (5−2)×180°=3×180°=540°(5-2) \times 180° = 3 \times 180° = 540°. Next, sum the four given angles: 90°+100°+105°+115°=410°90° + 100° + 105° + 115° = 410°. To find the fifth angle, subtract this sum from the total sum of the angles: 540°−410°=130°540° - 410° = 130°.

Question 3

The exterior angle of a regular polygon is 1/8 of its interior angle. How many sides does the polygon have?

  1. 8
  2. 9
  3. 16
  4. 18 (correct answer)
Explanation: Let the exterior angle be xx and the interior angle be yy. We are given x=18yx = \frac{1}{8}y, which means y=8xy = 8x. The interior and exterior angles at a vertex are supplementary, so x+y=180°x + y = 180°. Substitute 8x8x for yy: x+8x=180°x + 8x = 180°, which simplifies to 9x=180°9x = 180°. Solving for xx gives x=20°x = 20°. The number of sides nn of a regular polygon is 360°360° divided by the measure of one exterior angle. So, n=360/20=18n = 360 / 20 = 18. The polygon has 18 sides.

Question 4

In isosceles triangle ABC, side AB is congruent to side AC. A line parallel to base BC intersects side AB at D and side AC at E. If the measure of the vertex angle BAC is 30°, what is the measure of angle ADE?

  1. 30°
  2. 150°
  3. 105°
  4. 75° (correct answer)
Explanation: First, find the measure of the base angles in the large isosceles triangle ABC. The sum of angles is 180°. The sum of the two equal base angles (ABC and ACB) is 180°−30°=150°180° - 30° = 150°. Each base angle is 150°/2=75°150° / 2 = 75°. So, angle ABC = 75°. Because the line segment DE is parallel to the base BC, the angle ADE and the angle ABC are corresponding angles. Therefore, their measures are equal. Angle ADE = 75°.

Question 5

In triangle ABC, the measure of the exterior angle at vertex C is 110°. The measure of the interior angle at vertex B is (2x+10)°(2x + 10)°, and the measure of the interior angle at vertex A is (3x)°(3x)°. What is the measure of angle A?

  1. 20°
  2. 50°
  3. 60° (correct answer)
  4. 70°
Explanation: The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. In this case, the exterior angle at C is equal to the sum of the interior angles at A and B. So, 3x+(2x+10)=1103x + (2x + 10) = 110. Combine like terms: 5x+10=1105x + 10 = 110. Subtract 10 from both sides: 5x=1005x = 100. Divide by 5: x=20x = 20. The question asks for the measure of angle A, which is 3x3x. So, angle A = 3(20)=60°3(20) = 60°.

Question 6

Parallel lines ll and mm are intersected by a transversal line tt. The measure of an angle is given by (3x+10)°(3x + 10)°. The measure of its alternate interior angle is (5x−30)°(5x - 30)°. What is the measure of a consecutive interior angle to the first angle?

  1. 20°
  2. 70°
  3. 110° (correct answer)
  4. 120°
Explanation: Alternate interior angles are equal. So, set the expressions equal to each other: 3x+10=5x−303x + 10 = 5x - 30. Solve for xx: subtract 3x3x from both sides to get 10=2x−3010 = 2x - 30. Add 30 to both sides: 40=2x40 = 2x. Divide by 2: x=20x = 20. Now find the measure of the first angle: 3(20)+10=60+10=70°3(20) + 10 = 60 + 10 = 70°. Consecutive interior angles are supplementary, meaning they add up to 180°. So, the measure of the required angle is 180°−70°=110°180° - 70° = 110°.

Question 7

In triangle XYZ, the measure of angle X is 10 degrees greater than the measure of angle Y, and the measure of angle Z is twice the measure of angle Y. What is the measure of the largest angle in the triangle?

  1. 42.5°
  2. 52.5°
  3. 85° (correct answer)
  4. 95°
Explanation: Let the measure of angle Y be yy. Then the measure of angle X is y+10y + 10, and the measure of angle Z is 2y2y. The sum of the angles in a triangle is 180°. So, (y+10)+y+2y=180(y + 10) + y + 2y = 180. Combine like terms: 4y+10=1804y + 10 = 180. Subtract 10 from both sides: 4y=1704y = 170. Divide by 4: y=42.5°y = 42.5°. Now find the measures of all three angles: Angle Y = 42.5°; Angle X = 42.5° + 10° = 52.5°; Angle Z = 2 * 42.5° = 85°. The largest angle is angle Z, which is 85°.

Question 8

A four-sided plot of land is a convex quadrilateral. A surveyor measures three of the interior angles to be 85°, 92°, and 78°. To ensure the survey is accurate for a flat, four-sided figure, what must the measure of the fourth angle be?

  1. 105° (correct answer)
  2. 95°
  3. 115°
  4. 125°
Explanation: The sum of the interior angles of any convex quadrilateral (a four-sided polygon) is always 360°. To find the measure of the fourth angle, first sum the three known angles: 85°+92°+78°=255°85° + 92° + 78° = 255°. Then, subtract this sum from the total of 360°: 360°−255°=105°360° - 255° = 105°. The fourth angle must be 105°.

Question 9

In quadrilateral PQRS, consecutive angles measure 85°, 110°, and 95°. If the quadrilateral is cyclic (inscribed in a circle), what must be true about the fourth angle?

  1. It measures 70° and is opposite the 110° angle (correct answer)
  2. It measures 70° and is opposite the 85° angle
  3. It measures 85° and is opposite the 95° angle
  4. It measures 95° and is opposite the 85° angle
Explanation: In a cyclic quadrilateral, opposite angles are supplementary. The fourth angle must be 360° - (85° + 110° + 95°) = 70°. Since consecutive angles are given as 85°, 110°, 95°, the fourth angle (70°) is opposite to 110°. Choice B incorrectly identifies which angle is opposite. Choices C and D give incorrect angle measures that don't satisfy the angle sum property.

Question 10

The sum of the interior angles of a polygon is 1980°. How many sides does this polygon have?

  1. 11
  2. 13 (correct answer)
  3. 15
  4. 17
Explanation: The sum of interior angles of an n-sided polygon is (n-2) × 180°. Setting up the equation: (n-2) × 180° = 1980°. Dividing both sides by 180°: n-2 = 11, so n = 13. Choice A (11) is the value of (n-2). Choice C (15) results from dividing 1980 by 132 instead of 180. Choice D (17) results from adding 4 to 13 instead of 2.

Question 11

A regular octagon has an interior angle that measures x°. What is the measure of each exterior angle of this octagon?

  1. 45° (correct answer)
  2. 135°
  3. 225°
  4. 315°
Explanation: Each exterior angle of any regular polygon measures 360° ÷ n, where n is the number of sides. For an octagon, each exterior angle = 360° ÷ 8 = 45°. Choice B (135°) is the measure of each interior angle of a regular octagon. Choice C (225°) is 45° × 5, possibly from miscounting or confusion. Choice D (315°) is 360° - 45°, representing a reflex angle interpretation.

Question 12

The measures of the four interior angles of a quadrilateral are in the ratio 1:3:5:6. What is the measure of the smallest angle?

  1. 12°
  2. 24° (correct answer)
  3. 72°
  4. 144°
Explanation: The sum of the interior angles of a quadrilateral is 360°. Let the angles be xx, 3x3x, 5x5x, and 6x6x. Their sum is x+3x+5x+6x=15xx + 3x + 5x + 6x = 15x. Set this sum equal to 360°: 15x=360°15x = 360°. Divide by 15 to find x=24°x = 24°. The smallest angle is represented by xx, so its measure is 24°.

Question 13

Two angles are supplementary. One angle is 30 degrees less than twice the other angle. What is the measure of the smaller angle?

  1. 50°
  2. 60°
  3. 70° (correct answer)
  4. 110°
Explanation: Let the two angles be xx and yy. Since they are supplementary, x+y=180x + y = 180. One angle is 30 less than twice the other, so we can write y=2x−30y = 2x - 30. Substitute the second equation into the first: x+(2x−30)=180x + (2x - 30) = 180. Combine like terms: 3x−30=1803x - 30 = 180. Add 30 to both sides: 3x=2103x = 210. Divide by 3: x=70x = 70. Now find the other angle: y=180−70=110y = 180 - 70 = 110. The two angles are 70° and 110°. The smaller angle is 70°.

Question 14

If the number of sides of a regular polygon is doubled, what is the effect on the measure of each exterior angle?

  1. It is halved. (correct answer)
  2. It is doubled.
  3. It is quartered.
  4. It does not change.
Explanation: The measure of one exterior angle of a regular polygon with nn sides is given by the formula 360°/n360°/n. If the number of sides is doubled to 2n2n, the new exterior angle measure will be 360°/(2n)360°/(2n), which simplifies to 12×(360°/n)\frac{1}{2} \times (360°/n). This is exactly half of the original exterior angle measure. Therefore, the measure of each exterior angle is halved.

Question 15

Two lines intersect, forming four angles. One of the angles measures 42°. What is the sum of the measures of the two angles that are adjacent to this 42° angle?

  1. 138°
  2. 276° (correct answer)
  3. 318°
  4. 360°
Explanation: When two lines intersect, adjacent angles are supplementary (add up to 180°). The 42° angle has two adjacent angles. The measure of the first adjacent angle is 180°−42°=138°180° - 42° = 138°. The second adjacent angle is on the other side and is also supplementary to the 42° angle, so its measure is also 180°−42°=138°180° - 42° = 138°. The sum of the measures of these two adjacent angles is 138°+138°=276°138° + 138° = 276°.

Question 16

In a right triangle, the measures of the two acute angles are (7x+2)°(7x + 2)° and (3x−2)°(3x - 2)°. What is the measure of the smaller acute angle?

  1. 9°
  2. 25° (correct answer)
  3. 65°
  4. 90°
Explanation: The two acute angles in a right triangle are complementary, meaning their sum is 90°. Set up the equation: (7x+2)+(3x−2)=90(7x + 2) + (3x - 2) = 90. Combine like terms: 10x=9010x = 90. Solve for xx: x=9x = 9. Now substitute x=9x = 9 into both expressions to find the angle measures. The first angle is 7(9)+2=63+2=65°7(9) + 2 = 63 + 2 = 65°. The second angle is 3(9)−2=27−2=25°3(9) - 2 = 27 - 2 = 25°. The smaller of the two angles is 25°.

Question 17

Two parallel lines are cut by a transversal. If one of the alternate interior angles measures (3x + 15)° and its corresponding alternate interior angle measures (5x - 25)°, what is the value of x?

  1. 10
  2. 20 (correct answer)
  3. 25
  4. 35
Explanation: Alternate interior angles are equal when parallel lines are cut by a transversal. Setting (3x + 15) = (5x - 25), we get 15 + 25 = 5x - 3x, so 40 = 2x, giving x = 20. Choice A (10) results from solving 3x + 15 + 5x - 25 = 180 (treating them as supplementary). Choice C (25) comes from setting 3x + 15 = 90. Choice D (35) results from solving 3x + 15 = 5x - 25 incorrectly as 15 + 25 = 5x + 3x.

Question 18

A regular pentagon is inscribed in a circle. If two adjacent vertices of the pentagon are connected to the center of the circle, what is the measure of the angle formed at the center?

  1. 36°
  2. 108°
  3. 72° (correct answer)
  4. 144°
Explanation: When you see a regular polygon inscribed in a circle, you're dealing with central angles that divide the circle into equal parts. A regular pentagon has 5 equal sides and 5 equal vertices, so the circle is divided into 5 congruent central angles. Since a complete circle measures 360°, each central angle in a regular pentagon measures 360°5=72°\frac{360°}{5} = 72°. When you connect two adjacent vertices to the center, you're creating exactly one of these central angles. Let's examine why the other answers are incorrect. Choice A (36°) represents half of the correct central angle - this might come from incorrectly thinking about the relationship between central and inscribed angles, or from confusing this with interior angle calculations. Choice B (108°) is the measure of each interior angle of a regular pentagon, which is calculated using the formula (n−2)×180°n\frac{(n-2) \times 180°}{n}, but this isn't what the question asks for. Choice D (144°) equals two central angles (2×72°=144°2 \times 72° = 144°), which would be the angle if you connected vertices that are separated by one vertex, not adjacent ones. The correct answer is C (72°) because it represents one-fifth of the complete 360° circle. Strategy tip: For any regular n-sided polygon inscribed in a circle, the central angle between adjacent vertices is always 360°n\frac{360°}{n}. Memorize this formula - it's a reliable shortcut that works for any regular polygon, from triangles to dodecagons.

Question 19

In triangle ABC, the exterior angle at vertex C measures 118°. If angle A measures 42°, what is the measure of angle B?

  1. 34°
  2. 62°
  3. 76° (correct answer)
  4. 138°
Explanation: When you encounter a triangle problem involving an exterior angle, you're working with one of geometry's most useful relationships: an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. Here, the exterior angle at vertex C measures 118°, and angle A measures 42°. Since the exterior angle at C is formed by extending one side of the triangle, it equals the sum of the two remote interior angles (A and B). So: 118°=42°+angle B118° = 42° + \text{angle B} Solving for angle B: angle B=118°−42°=76°\text{angle B} = 118° - 42° = 76° Let's examine why the other answers miss the mark. Choice A (34°) would result if you mistakenly subtracted the exterior angle from angle A (42°−118°42° - 118°, then took the absolute value), which reverses the relationship. Choice B (62°) appears if you incorrectly assume the exterior angle plus angle A should equal 180° and solve 180°−118°=62°180° - 118° = 62°, but this confuses the exterior angle theorem with linear pair relationships. Choice D (138°) would make the sum of all interior angles exceed 180° (42°+138°=180°42° + 138° = 180°, leaving 0° for angle C), violating the triangle angle sum theorem. You can verify: if angle B = 76°, then angle C = 180°−42°−76°=62°180° - 42° - 76° = 62°, and the exterior angle should equal 42°+76°=118°42° + 76° = 118° ✓ Remember this key relationship: exterior angle = sum of the two remote interior angles. It's often faster than finding all three interior angles first.