Question

completing a two-column proof
statements\treasons

1. $\angle abc$ is rt. $\angle$\t1. a
2. $\overline{db}$ bisects $\angle abc$\t2. given
3. b\t3. def. of rt. $\angle$
4. $m\angle abd = m\angle cbd$\t4. def. of bis.
5. $m\angle abd + m\angle cbd = 90^\circ$\t5. c
6. $m\angle cbd + m\angle cbd = 90^\circ$\t6. subs. prop.
7. d\t7. add
8. $m\angle cbd = 45^\circ$\t8. div. prop.

identify the missing parts in the proof.
given: $\angle abc$ is a right angle.
$\overline{db}$ bisects $\angle abc$.
prove: $m\angle cbd = 45^\circ$
a: options: division, definition of bisects, definition of right angle, given
b: dropdown
c: dropdown
d: dropdown

Explanation:

Step1: Analyze Part A

The first statement is “\(\angle ABC\) is rt. \(\angle\)”, and the reason should be the given information. So A is "given".

Step2: Analyze Part B

The reason for statement 3 is “def. of rt. \(\angle\)”, which means the measure of a right angle is \(90^\circ\). So B is \(m\angle ABC = 90^\circ\).

Step3: Analyze Part C

Statement 5 is \(m\angle ABD + m\angle CBD = 90^\circ\), and this comes from the angle addition postulate (since \(\angle ABC=\angle ABD+\angle CBD\) and \(m\angle ABC = 90^\circ\)). So C is "angle addition postulate" (or from the fact that \(\angle ABC\) is a right angle, so the sum of its parts is \(90^\circ\)).

Step4: Analyze Part D

Statement 6 is \(m\angle CBD + m\angle CBD = 90^\circ\) (substitution from \(m\angle ABD = m\angle CBD\)), so statement 7 (with reason "add") should be \(2m\angle CBD = 90^\circ\) (adding \(m\angle CBD\) and \(m\angle CBD\)). So D is \(2m\angle CBD = 90^\circ\).

Answer:

A: given
B: \(m\angle ABC = 90^\circ\)
C: angle addition postulate (or related to \(\angle ABC\) being right angle)
D: \(2m\angle CBD = 90^\circ\)

(For the dropdowns: A is "given", B is "definition of right angle" (since \(m\angle ABC = 90^\circ\) is from def. of rt. \(\angle\)), C is from angle addition (or the fact that \(\angle ABC\) is right angle so sum of its parts is \(90^\circ\)), D is \(2m\angle CBD = 90^\circ\))