Question

writing a two-column proof
statements | reasons

1. ∠abc is rt. ∠ | 1. a
2. ( overline{db} ) bisects ( angle abc ) | 2. given
3. b | 3. def. of rt. ∠
4. ( mangle abd = mangle cbd ) | 4. def. of bis.
5. ( mangle abd + mangle cbd = 90^circ ) | 5. c
6. ( mangle cbd + mangle cbd = 90^circ ) | 6. subs. prop.
7. d | 7. add
8. ( mangle cbd = 45^circ ) | 8. div. prop.

identify the missing parts in the proof.
given: ( angle abc ) is a right angle.
( overline{db} ) bisects ( angle abc ).
prove: ( mangle cbd = 45^circ )
a: dropdown with \given\, b: dropdown with options including \measure of angle abc = 90\, c: ..., d: ...

Explanation:

Step1: Analyze A

The first statement is "$\angle ABC$ is rt. $\angle$", and the reason A should be "given" since it's the given information. So A is correct as "given".

Step2: Analyze B

The reason for statement 3 is "def. of rt. $\angle$", which means a right angle has a measure of $90^\circ$. So statement B should be "$m\angle ABC = 90^\circ$".

Step3: Analyze C

Statement 5 is "$m\angle ABD + m\angle CBD = 90^\circ$", and since $\angle ABC$ is $90^\circ$ and $\angle ABD + \angle CBD=\angle ABC$ (angle addition postulate), the reason C is "angle addition postulate" (or from the fact that $\angle ABC = 90^\circ$ and the two angles make it up). But for the option, we focus on the statement and reason matching.

Step4: Analyze D

From statement 6: $m\angle CBD + m\angle CBD = 90^\circ$ (which is $2m\angle CBD = 90^\circ$), then by the addition (combining like terms) we get that, and then dividing gives $m\angle CBD = 45^\circ$. But for statement 7, after combining $m\angle CBD + m\angle CBD$ we get $2m\angle CBD = 90^\circ$, so statement D should be "$2m\angle CBD = 90^\circ$". But for the given options for B, the correct one is "measure of angle ABC = 90".

Answer:

A: given
B: measure of angle ABC = 90
(For C, the reason is angle addition postulate; for D, the statement is $2m\angle CBD = 90^\circ$ and then division gives the final answer, but based on the options for B, the correct choice is as above.)