Question

Complete each sentence with the most appropriate word, phase or value.

$\overleftrightarrow{\mathit{\text{BD}}}$bisects $\angle \mathit{\text{ABC}}$,$\mathit{\text{m}}\angle \mathit{\text{ABC}}\text{=}5\mathit{\text{x}}-4$and $\mathit{\text{m}}\angle \mathit{\text{CBD}}\text{=}2\mathit{\text{x}}+10$. $\angle \mathit{\text{ABC}}$is a(n) ________ angle.

Short answer

The answer is Obtuse Angle.

Step-by-step solution

Step 1. State the property of bisector.  

There are two properties of a bisector:

1. Any point on the bisector of an angle is equidistant from the sides of the angle.

2. In a triangle, the angle bisector divides the opposite side in the ratio of the adjacent sides.

Step 2. State the explanation.

By using the property of the bisector,

$\begin{array}{c}m\angle ABC=2m\angle CBD\\ 5x-4=2(2x+10)\\ 5x-4=4x+20\\ 5x-4x=20+4\\ x=24\end{array}$

Then,

$\begin{array}{c}m\angle ABC=5\left(24\right)-4\\ =116\end{array}$

Since the measure of the angle is greater than 90 degrees but less than 180 degrees, $m\angle ABC$is an obtuse angle.

Hence, the blank is filled by the term “Obtuse Angle”.

Step 3. State the conclusion.

Therefore, “$\overrightarrow{BD}$bisects $\angle ABC$,$m\angle ABC=5x-4$ and $m\angle CBD=2x+10$.$\angle ABC$ is a(n) obtuse angle”.