Question

writing a two - column proof
statements | reasons

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

identify the missing parts in the proof.
given: ∠abc is a right angle, $overline{db}$ bisects ∠abc.
prove: $m∠cbd = 45°$
a:

b:

c:

d:

Explanation:

Step1: Determine A (Reason for ∠ABC is rt ∠)

The reason for stating ∠ABC is a right angle is the "given" information (since it's provided as a given in the problem). Wait, no—wait, the "def of rt ∠" is for the measure. Wait, the first statement is ∠ABC is rt ∠, so the reason (A) should be "given" (since the problem states "Given: ∠ABC is a right angle"). Wait, but the options—wait, maybe I misread. Wait, the "Reasons" column: 1. A, and the given is ∠ABC is right angle. So A is "given".

Step2: Determine B (Statement 3)

The definition of a right angle (def of rt ∠) means that \( m\angle ABC = 90^\circ \). So statement B should be \( m\angle ABC = 90^\circ \).

Step3: Determine C (Reason for \( m\angle ABD + m\angle CBD = 90^\circ \))

Since \( \angle ABC \) is a right angle (so \( m\angle ABC = 90^\circ \)) and \( \angle ABD + \angle CBD = \angle ABC \) (angle addition postulate), the reason C is "angle addition postulate" (or "def of angle addition").

Step4: Determine D (Statement 7)

After substituting \( m\angle ABD = m\angle CBD \) into \( m\angle ABD + m\angle CBD = 90^\circ \), we get \( m\angle CBD + m\angle CBD = 90^\circ \), which simplifies to \( 2m\angle CBD = 90^\circ \) (by addition, combining like terms). So statement D is \( 2m\angle CBD = 90^\circ \).

Answer:

A: given
B: \( m\angle ABC = 90^\circ \)
C: angle addition postulate (or def of angle addition)
D: \( 2m\angle CBD = 90^\circ \)