Question

using the following diagram, find the indicated angle measure.
question 8 (1 point)
given that $overline{bd}$ bisects $angle abc$, find $mangle abc$.

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. Angle \(b\) and the \(74^{\circ}\) angle are vertical angles. So \(b = 74^{\circ}\).

Step2: Use angle - bisector property for \(\angle ABC\)

Since \(\overline{BD}\) bisects \(\angle ABC\), then \(\angle ABD=\angle DBC\). So \(5x + 16=8x-23\).
Solve for \(x\):
\[

$$\begin{align*} 8x-5x&=16 + 23\\ 3x&=39\\ x& = 13 \end{align*}$$

\]

Step3: Find \(\angle ABD\) or \(\angle DBC\)

Substitute \(x = 13\) into \(\angle ABD=5x + 16\). Then \(\angle ABD=5\times13+16=65 + 16=81^{\circ}\).

Step4: Calculate \(\angle ABC\)

Since \(\angle ABC = 2\angle ABD\) (because of angle - bisector), \(\angle ABC=2\times81^{\circ}=162^{\circ}\).

Answer:

Blank 1 for first question: \(74\)
Blank 1 for second question: \(162\)