Question

if ur bisects ∠tuv, m∠tuv=(13x - 5)° and m∠ruv=(7x + 31)°, find the value of ∠tuv.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{UR}$ bisects $\angle TUV$, then $m\angle TUR=m\angle RUV$. So, $13x - 5=2(7x + 31)$.

Step2: Solve the equation for $x$

Expand the right - hand side: $13x-5 = 14x+62$.
Subtract $13x$ from both sides: $-5=x + 62$.
Subtract 62 from both sides: $x=-67$.

Step3: Calculate the measure of $\angle TUV$

Substitute $x = - 67$ into the expression for $m\angle TUV=13x - 5$.
$m\angle TUV=13\times(-67)-5=-871 - 5=-876$. But angle measure cannot be negative. There is a mistake above. The correct equation should be $13x-5 = 2(7x + 31)$ is wrong. It should be $13x-5=2(7x + 31)$ is wrong. Since $\overrightarrow{UR}$ bisects $\angle TUV$, we have $m\angle TUR = m\angle RUV$, so $13x-5=7x + 31$.
Subtract $7x$ from both sides: $13x-7x-5=7x-7x + 31$, $6x-5 = 31$.
Add 5 to both sides: $6x-5 + 5=31 + 5$, $6x=36$.
Divide both sides by 6: $x = 6$.

Step4: Find the measure of $\angle TUV$

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

Answer:

$73^{\circ}$