Question

3 fill in the blank 2 points
$overrightarrow{bd}$ bisects $angle abc$. what is the value of $x$? what is $mangle abc$?
diagram: point b with rays ba, bd, bc; angle between ba and bd is $(x + 15)^circ$, between bd and bc is $(4x - 45)^circ$
$x =$ type your answer...
$mangle abc =$ type your answer...°

Explanation:

Step1: Use Angle Bisector Definition

Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), then \(\angle ABD=\angle DBC\). So we set up the equation:
\(x + 15=4x - 45\)

Step2: Solve for \(x\)

Subtract \(x\) from both sides:
\(15 = 3x - 45\)
Add 45 to both sides:
\(60 = 3x\)
Divide both sides by 3:
\(x = 20\)

Step3: Find \(m\angle ABC\)

First, find \(m\angle ABD\) and \(m\angle DBC\) by substituting \(x = 20\).
\(m\angle ABD=(20 + 15)^\circ=35^\circ\)
\(m\angle DBC=(4\times20 - 45)^\circ=(80 - 45)^\circ = 35^\circ\)
Then, \(m\angle ABC=m\angle ABD + m\angle DBC=35^\circ+35^\circ = 70^\circ\)

Answer:

\(x = 20\)
\(m\angle ABC = 70\)