Question

1. if (overline{uw}) bisects (angle tuv), (mangle tuw=(13x - 5)^{circ}) and (mangle wuv=(7x + 31)^{circ}), find the value of (x).

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{UW}$ bisects $\angle TUV$, then $m\angle TUW=m\angle WUV$. So, $13x - 5=7x + 31$.

Step2: Solve the equation for x

Subtract $7x$ from both sides: $(13x-7x)-5=(7x - 7x)+31$, which simplifies to $6x-5 = 31$.
Then add 5 to both sides: $6x-5 + 5=31 + 5$, giving $6x=36$.
Divide both sides by 6: $\frac{6x}{6}=\frac{36}{6}$, so $x = 6$.

Step3: Find $m\angle TUW$

Substitute $x = 6$ into the expression for $m\angle TUW$. $m\angle TUW=13x-5=13\times6-5=78 - 5=73^{\circ}$.

Step4: Find $m\angle TUV$

Since $m\angle TUV=m\angle TUW + m\angle WUV$ and $m\angle TUW=m\angle WUV = 73^{\circ}$, then $m\angle TUV=2\times73^{\circ}=146^{\circ}$.

Answer:

$x = 6$, $m\angle TUV=146^{\circ}$