Question

writing a two - column proof

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 \\( \checkmark \\) given
b
c
d measure of angle \\( abc = 90 \\) \\( cbd = 90 \\)
measure of angle \\( abc = 180 \\)
x measure of angle \\( abd = 90 \\)
measure of angle \\( cbd = 180 \\)

Explanation:

Step1: Analyze statement 3

Since the reason is “def. of rt. \(\angle\)” (definition of a right angle), and a right - angle has a measure of \(90^{\circ}\). So, statement 3 should be \(m\angle ABC = 90^{\circ}\) (because \(\angle ABC\) is a right angle).

Step2: Analyze reason 5

The statement \(m\angle ABD + m\angle CBD=90^{\circ}\) is based on the angle - addition postulate. The angle - addition postulate states that if \(D\) is in the interior of \(\angle ABC\), then \(m\angle ABD + m\angle DBC=m\angle ABC\). Here, \(\angle ABC\) is a right angle (\(m\angle ABC = 90^{\circ}\)), so \(m\angle ABD + m\angle CBD=90^{\circ}\) by the angle - addition postulate.

Step3: Analyze statement 7

From \(m\angle CBD + m\angle CBD=90^{\circ}\) (statement 6), using the addition property of equality (combining like terms), we get \(2m\angle CBD = 90^{\circ}\) (statement 7). Then, by the division property of equality (dividing both sides by 2), we get \(m\angle CBD=45^{\circ}\) (statement 8).

Answer:

B. \(m\angle ABC = 90^{\circ}\)
C. angle - addition postulate
D. \(2m\angle CBD=90^{\circ}\)