Question

1. if $overrightarrow{uw}$ bisects $angle tuv$, $mangle tuw=(13x - 5)^{circ}$ and $mangle wuv=(7x + 31)^{circ}$, find the value of $x$.
2. if $overrightarrow{mo}$ bisects $angle pmn$, $mangle pmn = 74^{circ}$ and $mangle omn=(2x + 7)^{circ}$, find the value of $x$.
3. if $overrightarrow{ef}$ bisects $angle ceb$, $mangle cef=(7x + 21)^{circ}$ and $mangle feb=(10x - 3)^{circ}$, find the measure of $angle deb$.

Explanation:

Step1: Use angle - bisector property

Since a ray that bisects an angle divides the angle into two equal - measure angles. For the first problem, if $\overrightarrow{UW}$ bisects $\angle TUV$, then $m\angle TUW=m\angle WUV$. So, we set up the equation $13x - 5=7x + 31$.
$13x-7x=31 + 5$
$6x=36$
$x = 6$

Step2: For the second problem

If $\overrightarrow{MO}$ bisects $\angle PMN$, then $m\angle PMN = 2m\angle OMN$. Given $m\angle PMN = 74^{\circ}$ and $m\angle OMN=(2x + 7)^{\circ}$, we have the equation $74 = 2(2x + 7)$.
First, distribute the 2: $74=4x+14$.
Then, subtract 14 from both sides: $4x=74 - 14=60$.
Divide both sides by 4: $x = 15$.

Step3: For the third problem

If $\overrightarrow{EF}$ bisects $\angle CEB$, then $m\angle CEF=m\angle FEB$. Set up the equation $7x + 21=10x-3$.
Subtract $7x$ from both sides: $21 = 10x-7x-3$.
$21=3x - 3$.
Add 3 to both sides: $3x=21 + 3=24$.
Divide both sides by 3: $x = 8$.
Then, $m\angle FEB=10x-3=10\times8 - 3=77^{\circ}$. But we need to find $m\angle DEB$. Since there is no information about the relationship between $\angle DEB$ and $\angle FEB$ in the problem - statement (assuming $\angle DEB$ and $\angle FEB$ are vertical angles or supplementary angles etc. is not given), if we assume $\angle DEB$ and $\angle FEB$ are vertical angles, then $m\angle DEB=m\angle FEB = 77^{\circ}$.

Answer:

1. $x = 6$
2. $x = 15$
3. $x = 8$, $m\angle DEB = 77^{\circ}$