Question

1. in the diagram below, $overrightarrow{bd}$ bisects $angle abc$. find the value of x and then find $mangle abc$. a (5x - 11)° d (4x + 1)° b c

Explanation:

Step1: Set up equation from angle - bisector property

Since $\overrightarrow{BD}$ bisects $\angle ABC$, then $\angle ABD=\angle DBC$. So, $5x - 11=4x + 1$.

Step2: Solve for x

Subtract $4x$ from both sides of the equation: $5x-4x-11=4x - 4x+1$, which simplifies to $x-11 = 1$. Then add 11 to both sides: $x=1 + 11$, so $x = 12$.

Step3: Find $m\angle ABC$

First, find $\angle ABD$ or $\angle DBC$ by substituting $x = 12$ into either expression. Let's use $\angle DBC=4x + 1$. Then $\angle DBC=4\times12+1=48 + 1=49^{\circ}$. Since $\angle ABC = 2\angle DBC$ (because of the angle - bisector), $m\angle ABC=2\times49^{\circ}=98^{\circ}$.

Answer:

$x = 12$, $m\angle ABC=98^{\circ}$