Question

part a
in the figure, \\(\overrightarrow{bd}\\) bisects \\(\angle abc\\).
find x and the \\(m\angle abc\\).
x =
part b
\\(m\angle abc=\\)

Explanation:

Step1: Use angle - bisector property

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

Step2: Solve the equation for $x$

Subtract $x$ from both sides: $15 = 3x-45$. Then add 45 to both sides: $60=3x$. Divide both sides by 3, we get $x = 20$.

Step3: Find the measure of $\angle ABC$

First find $\angle ABD$ or $\angle DBC$ by substituting $x = 20$ into either expression. $\angle ABD=x + 15=20+15 = 35^{\circ}$. Since $\angle ABC=2\angle ABD$, then $\angle ABC=2\times35^{\circ}=70^{\circ}$.

Answer:

Part A: $x = 20$
Part B: $m\angle ABC=70^{\circ}$