Question

1. find m∠tuv if m∠tug = 34°, m∠tuv=x + 164, and m∠guv = 2x+130.
2. find m∠cmn if m∠lmn = 108°, m∠cmn=x + 53, and m∠lmc=x + 65.
3. find m∠deq if m∠dep = 148°, m∠deq=x + 42, and m∠qep=x + 124.
4. find m∠whg if m∠whg = 17x+5, m∠jhw = 9x + 11, and m∠jhg = 172°.
5. m∠rst = 102°, m∠rsb = 16x, and m∠bst = 9x+2. find m∠bst.
6. find m∠lkd if m∠lkd=x + 60, m∠lkj = 171°, and m∠dkj=x + 129.
7. m∠rql = 7x+8, m∠lqp = 4x - 11, and m∠rqp = 118°. find m∠lqp.
8. m∠tuh = 2x - 2, m∠huv = 92°, and m∠tuv = 13+9x. find m∠tuh.
9. find m∠kjl if m∠kjl=x + 34, m∠lji=x + 68, and m∠kji = 86°.
10. m∠vkj = 120°, m∠lkv = 3x - 3, and m∠lkj = 17x - 9. find m∠lkv.

Explanation:

Step1: Use angle - addition postulate

The angle - addition postulate states that if a point lies on the interior of an angle, the measure of the whole angle is equal to the sum of the measures of the two smaller angles. For example, if $\angle ABC$ is composed of $\angle ABD$ and $\angle DBC$, then $m\angle ABC=m\angle ABD + m\angle DBC$.

Let's take problem 9 as an example:
We know that $m\angle TUV=m\angle TUG + m\angle GUV$. Given $m\angle TUG = 34^{\circ}$, $m\angle TUV=x + 164$, and $m\angle GUV=2x + 130$.
So, we set up the equation:
$x + 164=34+(2x + 130)$

Step2: Solve the equation for x

First, simplify the right - hand side of the equation:
$x + 164=2x+164$
Subtract $x$ from both sides:
$x - x+164=2x - x + 164$
$0=x$

Then find $m\angle TUV$:
Since $x = 0$, $m\angle TUV=x + 164=164^{\circ}$

We can follow the same steps for other problems:

For problem 10:
We know that $m\angle LMN=m\angle LMC + m\angle CMN$. Given $m\angle LMN = 108^{\circ}$, $m\angle CMN=x + 53$, and $m\angle LMC=x + 65$.
Set up the equation: $108=(x + 65)+(x + 53)$
Simplify the right - hand side: $108=2x+118$
Subtract 118 from both sides: $108 - 118=2x$, so $- 10=2x$
Divide both sides by 2: $x=-5$
$m\angle CMN=x + 53=-5 + 53 = 48^{\circ}$

Answer:

1. $m\angle TUV = 164^{\circ}$
2. $m\angle CMN = 48^{\circ}$
3. First, $m\angle DEP=m\angle DEQ + m\angle QEP$. Given $m\angle DEP = 148^{\circ}$, $m\angle DEQ=x + 42$, and $m\angle QEP=x + 124$.

Set up the equation: $148=(x + 42)+(x + 124)$
Simplify: $148=2x + 166$
Subtract 166 from both sides: $148-166=2x$, $-18 = 2x$
$x=-9$
$m\angle DEQ=x + 42=-9 + 42=33^{\circ}$

1. First, $m\angle WHG=m\angle WHI + m\angle IHG$. Given $m\angle WHG=17x + 5$, $m\angle WHI=9x + 11$, and $m\angle IHG = 172^{\circ}$.

Set up the equation: $17x + 5=(9x + 11)+172$
Simplify: $17x + 5=9x+183$
Subtract $9x$ from both sides: $17x-9x + 5=9x-9x + 183$
$8x+5=183$
Subtract 5 from both sides: $8x=178$
$x=\frac{89}{4}$
$m\angle WHG=17\times\frac{89}{4}+5=\frac{1513 + 20}{4}=\frac{1533}{4}=383.25^{\circ}$ (There may be a mistake in the problem - setup as angles should be between $0^{\circ}$ and $360^{\circ}$)

1. First, $m\angle RST=m\angle RSB + m\angle BST$. Given $m\angle RST = 102^{\circ}$, $m\angle RSB = 16x$, and $m\angle BST=9x + 2$.

Set up the equation: $102=16x+(9x + 2)$
Simplify: $102=25x+2$
Subtract 2 from both sides: $100=25x$
$x = 4$
$m\angle BST=9x + 2=9\times4+2=38^{\circ}$

1. First, $m\angle LKJ=m\angle LKD + m\angle DKJ$. Given $m\angle LKJ = 171^{\circ}$, $m\angle LKD=x + 60$, and $m\angle DKJ=x + 129$.

Set up the equation: $171=(x + 60)+(x + 129)$
Simplify: $171=2x+189$
Subtract 189 from both sides: $171-189=2x$
$-18=2x$
$x=-9$
$m\angle LKD=x + 60=-9 + 60 = 51^{\circ}$

1. First, $m\angle RQP=m\angle RQL + m\angle LQP$. Given $m\angle RQP = 118^{\circ}$, $m\angle RQL=7x + 8$, and $m\angle LQP=4x-11$.

Set up the equation: $118=(7x + 8)+(4x-11)$
Simplify: $118=11x-3$
Add 3 to both sides: $121=11x$
$x = 11$
$m\angle LQP=4x-11=4\times11-11=33^{\circ}$

1. First, $m\angle TUV=m\angle TUH + m\angle HUV$. Given $m\angle TUV=13 + 9x$, $m\angle TUH=2x-2$, and $m\angle HUV = 92^{\circ}$.

Set up the equation: $13 + 9x=(2x-2)+92$
Simplify: $13 + 9x=2x+90$
Subtract $2x$ from both sides: $13 + 9x-2x=2x-2x+90$
$13 + 7x=90$
Subtract 13 from both sides: $7x=77$
$x = 11$
$m\angle TUH=2x-2=2\times11-2=20^{\circ}$

1. First, $m\angle KJI=m\angle KJL + m\angle LJI$. Given $m\angle KJI = 86^{\circ}$, $m\angle KJL=x + 34$, and $m\angle LJI=x + 68$.

Set up the equation: $86=(x + 34)+(x + 68)$
Simplify: $86=2x+102$
Subtract 102 from both sides: $86 - 102=2x$
$-16=2x$
$x=-8$
$m\angle KJL=x + 34=-8 + 34 = 26^{\circ}$

1. First, $m\angle VKJ=m\angle LKV + m\angle LKJ$. Given $m\angle VKJ = 120^{\circ}$, $m\angle LKV=3x-3$, and $m\angle LKJ=17x-9$.

Set up the equation: $120=(3x-3)+(17x-9)$
Simplify: $120=20x-12$
Add 12 to both sides: $132=20x$
$x=\frac{33}{5}$
$m\angle LKV=3x-3=3\times\frac{33}{5}-3=\frac{99 - 15}{5}=\frac{84}{5}=16.8^{\circ}$