📐 Geometry · Basics

Angle Relationships practice questions

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Answer key with explanations

1. What is the complement of a 76° angle?

   Answer: 14°. Complementary angles add to 90°, so the complement of 76° is 90° − 76° = 14°. Check: 76° + 14° = 90°. The choice 104° is the supplement (180° − 76°). The choice 166° adds instead of subtracting (90° + 76°). The choice 24° is a slip: it is 100° − 76°.

2. Two lines cross. One of the four angles measures 107°. What does the angle directly across from it (its vertical angle) measure?

   Answer: 107°. Vertical angles are equal, so the angle directly across from a 107° angle also measures 107°. The choice 73° is each neighbor (180° − 107°, from a linear pair). The choice 253° is the total of the other three angles (107° + 73° + 73°). The choice 17° is 107° − 90°, which has no meaning here.

3. Which pair of angle measures is supplementary?

   Answer: 47° and 133°. Supplementary angles add to 180°, and 47° + 133° = 180°. The others do not: 62° + 28° = 90° (complementary), 110° + 80° = 190°, and 95° + 75° = 170°.

4. The angles (4x + 7)° and (2x + 5)° form a linear pair. What is x?

   Answer: 28. A linear pair adds to 180°: (4x + 7) + (2x + 5) = 180, so 6x + 12 = 180, 6x = 168 and x = 28. Check: the angles are 4(28) + 7 = 119° and 2(28) + 5 = 61°, and 119° + 61° = 180°. The choice 13 comes from using 90° instead of 180°. The choice 30 forgets the 12 (6x = 180). The choice −1 sets the two angles equal, as if they were vertical angles.

5. The angles (8x − 5)° and (5x + 16)° are vertical angles. What is the measure of each angle?

   Answer: 51°. Vertical angles are equal, so 8x − 5 = 5x + 16. Subtract 5x: 3x − 5 = 16. Add 5: 3x = 21, so x = 7. Each angle is 8(7) − 5 = 51° and 5(7) + 16 = 51°. The choice 7° is x, not the angle. The choice 129° is the neighboring angle (180° − 51°). The choice 102° is 51° + 51°.

6. Ray BD bisects ∠ABC. m∠ABD = (3x + 4)° and m∠DBC = (5x − 18)°. What is m∠ABC?

   Answer: 74°. A bisector makes two equal angles, so 3x + 4 = 5x − 18. Subtract 3x: 4 = 2x − 18. Add 18: 22 = 2x, so x = 11. Each half is 3(11) + 4 = 37° and 5(11) − 18 = 37°. The whole angle is m∠ABC = 37° + 37° = 74°. The choice 37° is only one half. The choice 11° is x. The choice 148° doubles the answer twice.

7. Three angles fill the space around a point with no gaps or overlaps. Their measures are x°, (x + 40)° and (2x − 20)°. What is the largest angle?

   Answer: 150°. Angles around a point add to 360°: x + (x + 40) + (2x − 20) = 360, so 4x + 20 = 360, 4x = 340 and x = 85. The angles are 85°, 85 + 40 = 125° and 2(85) − 20 = 150°, which add to 360°. The largest is 150°. The choice 85° is x, the smallest angle. The choice 125° is the middle angle. The choice 80° comes from using 180° instead of 360° (that gives x = 40 and a largest angle of 80°).

8. Point P lies on line AC, between A and C, and B is a point not on that line. Nothing else is marked: no measures, no equal-angle marks, no right-angle square. In the picture, ∠APB and ∠BPC look equal. Which statement is justified by what is given?

   Answer: m∠APB + m∠BPC = 180°. Because P is between A and C on a line, rays PA and PC are opposite rays, so ∠APB and ∠BPC form a linear pair, and their measures add to 180°. Nothing marks the angles as equal or as right angles, and diagrams are not drawn to scale, so you cannot assume either. They cannot be complementary, because a linear pair adds to 180°, not 90°. For example, 123° and 57° fit everything that is given.

9. An angle measures 21° more than twice its complement. What is the angle?

   Answer: 67°. Let x be the angle. Its complement is 90 − x. The angle is 21 more than twice its complement, so x = 2(90 − x) + 21. Distribute: x = 180 − 2x + 21, so x = 201 − 2x. Add 2x: 3x = 201, so x = 67. Check: the complement is 90° − 67° = 23°, and 2(23°) + 21° = 67°. The choice 23° is the complement. The choice 127° uses the supplement by mistake (x = 2(180 − x) + 21). The choice 53° subtracts the 21 instead of adding it (x = 2(90 − x) − 21).

10. Two lines cross. One angle measures (x + 25)°. The angle next to it along a straight line measures (4x − 15)° (the two form a linear pair). What is the measure of the angle vertical to the (x + 25)° angle?

    Answer: 59°. A linear pair adds to 180°: (x + 25) + (4x − 15) = 180, so 5x + 10 = 180, 5x = 170 and x = 34. The angle (x + 25)° measures 34 + 25 = 59°, and its neighbor measures 4(34) − 15 = 121°. A vertical angle equals the angle directly across from it, so the answer is 59°. The choice 121° is the neighbor. The choice 34° is x. The choice 146° is 180° − x.