Question

1. complete the proof.

given: \\( \angle a d b \\) and \\( \angle b d c \\) form a linear pair and \\( \angle a d b \cong \angle b d c \\).
prove: \\( \angle a d b \\) and \\( \angle b d c \\) are right angles.

1. \\( \angle a d b \\) and \\( \angle b d c \\) form a linear pair and \\( \angle a d b \cong \angle b d c \\).
2. \\( \angle a d b \\) and \\( \angle b d c \\) are a complementary b supplementary.
3. \\( m \angle a d b + m \angle b d c = 180 ^ { \circ } \\)
4. \\( m \angle a d b = m \angle b d c \\)
5. \\( m \angle a d b + m \angle a d b = 180 ^ { \circ } \\)
6. \\( 2 m \angle a d b = 180 ^ { \circ } \\)
7. \\( m \angle a d b = 90 ^ { \circ } \\)
8. \\( m \angle d b c = 90 ^ { \circ } \\)
9. \\( \angle a d b \\) and \\( \angle d b c \\) are right angles.
10. given
11. linear pair theorem
12. a definition of linear pair b definition of supplementary angles
13. definition of congruence
14. substitution
15. substitution
16. division property of equality
17. congruent angles have the same measure.
18. a definition of congruence b definition of right angles

Explanation:

Step1: Complete Statement 2

According to the Linear Pair Theorem, a linear pair of angles is supplementary. So, for statement 2, the answer is supplementary.

Step2: Complete Statement 3

Since \(\angle ADB\) and \(\angle BDC\) are supplementary (from statement 2), by the definition of supplementary angles, \(m\angle ADB + m\angle BDC=180^{\circ}\). So, for statement 3, the answer is definition of supplementary angles.

Step3: Complete Statement 9

Since \(m\angle ADB = 90^{\circ}\) and \(m\angle BDC = 90^{\circ}\), by the definition of right - angles (an angle with a measure of \(90^{\circ}\) is a right - angle), \(\angle ADB\) and \(\angle BDC\) are right angles. So, for statement 9, the answer is definition of right angles.

Answer:

1. B. supplementary
2. B. Definition of supplementary angles
3. B. Definition of right angles