Question

decide whether the statement is true or false. if the statement is false, reword the statement so that the statement is true. 1. two angles are complementary if the sum of their measures is 180°. 2. two angles are supplementary if the sum of their measures is 180°. 3. two angles are adjacent angles if they share a common vertex. 4. a theorem is a true statement that follows from other true statements. determine whether the angles are complementary, supplementary, or neither. 5. image of angles with measures 60° and 30° 6. image of angles with measures 30° and 150° 7. image of angles with measures 37° and 55° find the measure of a complement of the angle given. 8. image of an angle with measure 32° 9. image of an angle with measure 15° 10. image of an angle with measure 77° find the measure of a supplement of the angle given. 11. image of an angle with measure 160° 12. image of a right - angled figure 13. image of an angle with measure 18° use the diagram to complete the statement. 14. ∠1 and? are supplementary angles. 15. ∠3 and? are supplementary angles. 16.? ≅? by the congruent supplements theorem.

Explanation:

Step1: Recall angle - relationship definitions

Complementary angles have a sum of 90°, supplementary angles have a sum of 180°, and adjacent angles share a common vertex and a common side.

Step2: Analyze statement 1

The statement "Two angles are complementary if the sum of their measures is 180°" is false. Complementary angles have a sum of 90°. The corrected statement is "Two angles are complementary if the sum of their measures is 90°".

Step3: Analyze statement 2

The statement "Two angles are supplementary if the sum of their measures is 180°" is true.

Step4: Analyze statement 3

The statement "Two angles are adjacent angles if they share a common vertex" is false. Adjacent angles must also share a common side. The corrected statement is "Two angles are adjacent angles if they share a common vertex and a common side".

Step5: Analyze statement 4

The statement "A theorem is a true statement that follows from other true statements" is true.

Step6: Analyze angles in 5 - 7

For 5: 30°+60° = 90°, so the angles are complementary.
For 6: 30°+150° = 180°, so the angles are supplementary.
For 7: 37°+55°=92°≠90° and 92°≠180°, so the angles are neither complementary nor supplementary.

Step7: Find complementary angles in 8 - 10

Complementary angle of 32° is 90° - 32°=58°.
Complementary angle of 15° is 90° - 15° = 75°.
Complementary angle of 77° is 90° - 77°=13°.

Step8: Find supplementary angles in 11 - 13

Supplementary angle of 160° is 180° - 160° = 20°.
Supplementary angle of 90° is 180° - 90°=90°.
Supplementary angle of 18° is 180° - 18° = 162°.

Step9: Analyze 14 - 16

In the intersection - of - lines diagram, ∠1 and ∠2 are supplementary angles (linear pair), ∠3 and ∠2 are supplementary angles (linear pair). By the Congruent Supplements Theorem, ∠1≅∠3.

Answer:

1. False. Two angles are complementary if the sum of their measures is 90°.
2. True.
3. False. Two angles are adjacent angles if they share a common vertex and a common side.
4. True.
5. Complementary.
6. Supplementary.
7. Neither.
8. 58°.
9. 75°.
10. 13°.
11. 20°.
12. 90°.
13. 162°.
14. ∠2.
15. ∠2.
16. ∠1≅∠3.