Question

1. find (mangle tuv) if (mangle tug = 34^{circ}), (mangle tuv=x + 164), and (mangle guv=2x + 130).
2. find (mangle cmn) if (mangle lmn = 108^{circ}), (mangle cmn=x + 53), and (mangle lmc=x + 65).
3. find (mangle deq) if (mangle def = 148^{circ}), (mangle deq=x + 42), and (mangle qef=x + 124).
4. find (mangle whg) if (mangle whg=17x + 5), (mangle ihw=9x + 11), and (mangle ihg = 172^{circ}).
5. (mangle rst = 102^{circ}), (mangle rss=16x), and (mangle sst = 9x+2). find (mangle sst).
6. find (mangle lkd) if (mangle lkd=x + 60), (mangle lkj = 171^{circ}), and (mangle dkj=x + 129).
7. (mangle rql=7x + 8), (mangle lqp=4x - 11), and (mangle rqp = 118^{circ}). find (mangle lqp).
8. (mangle tuh=2x - 2), (mangle huv = 92^{circ}), and (mangle tuv = 13+9x). find (mangle tuh).
9. find (mangle kjl) if (mangle kjl=x + 34), (mangle lji=x + 68), and (mangle kji = 86^{circ}).
10. (mangle vkj = 120^{circ}), (mangle lkv=3x - 3), and (mangle lkj=17x - 9). find (mangle lkv).

Explanation:

Step1: Identify angle - addition relationships

In each problem, we use the fact that the measure of a larger angle is the sum of the measures of its non - overlapping sub - angles. For example, if \(\angle ABC=\angle ABD+\angle DBC\), we can set up an equation based on the given angle measures.

Step2: Solve for the variable \(x\)

For problem 9:
We know that \(m\angle TUV=m\angle TUG + m\angle GUV\). So, \(x + 164=34+(2x + 130)\).
First, simplify the right - hand side: \(x + 164=2x+164\).
Subtract \(x\) from both sides: \(164=x + 164\), then \(x = 0\).
For problem 10:
Since \(m\angle LMN=m\angle LMC+m\angle CMN\), we have \(108=(x + 65)+(x + 53)\).
Combine like terms: \(108 = 2x+118\).
Subtract 118 from both sides: \(2x=108 - 118=-10\), then \(x=-5\).
For problem 11:
As \(m\angle DEF=m\angle DEQ+m\angle QEF\), we get \(148=(x + 42)+(x + 124)\).
Combine like terms: \(148=2x + 166\).
Subtract 166 from both sides: \(2x=148 - 166=-18\), then \(x=-9\).
For problem 12:
Since \(m\angle WHG=m\angle WHN+m\angle NHG\), we have \(17x + 5=(9x + 11)+172\).
Combine like terms: \(17x+5=9x + 183\).
Subtract \(9x\) from both sides: \(8x+5=183\).
Subtract 5 from both sides: \(8x=178\), then \(x=\frac{89}{4}=22.25\).
For problem 13:
Given \(m\angle RST=m\angle RSB+m\angle BST\), so \(102=16x+(9x + 2)\).
Combine like terms: \(102=25x + 2\).
Subtract 2 from both sides: \(25x=100\), then \(x = 4\). And \(m\angle BST=9x + 2=9\times4+2=38^{\circ}\).
For problem 14:
Since \(m\angle LKJ=m\angle LKD+m\angle DKJ\), we have \(171=(x + 60)+(x + 129)\).
Combine like terms: \(171=2x+189\).
Subtract 189 from both sides: \(2x=171 - 189=-18\), then \(x=-9\).
For problem 15:
As \(m\angle RQP=m\angle RQL+m\angle LQP\), we get \(118=(7x + 8)+(4x - 11)\).
Combine like terms: \(118=11x - 3\).
Add 3 to both sides: \(11x=121\), then \(x = 11\). And \(m\angle LQP=4x-11=4\times11-11 = 33^{\circ}\).
For problem 16:
Since \(m\angle TUV=m\angle TUH+m\angle HUV\), we have \(13 + 9x=(2x - 2)+92\).
Combine like terms: \(13 + 9x=2x+90\).
Subtract \(2x\) from both sides: \(7x+13=90\).
Subtract 13 from both sides: \(7x=77\), then \(x = 11\). And \(m\angle TUH=2x-2=2\times11-2 = 20^{\circ}\).
For problem 17:
Given \(m\angle KJL=m\angle KJI+m\angle IJL\), so \(x + 34=(x + 68)+86\).
This gives \(x + 34=x + 154\), which has no solution as \(34
eq154\).
For problem 18:
Since \(m\angle VKJ=m\angle VKL+m\angle LKJ\), we have \(120=(3x - 3)+(17x - 9)\).
Combine like terms: \(120=20x-12\).
Add 12 to both sides: \(20x=132\), then \(x=\frac{33}{5}=6.6\). And \(m\angle LKV=3x - 3=3\times6.6-3=16.8^{\circ}\).

Answer:

1. \(x = 0\)
2. \(x=-5\)
3. \(x=-9\)
4. \(x = 22.25\)
5. \(x = 4\), \(m\angle BST = 38^{\circ}\)
6. \(x=-9\)
7. \(x = 11\), \(m\angle LQP=33^{\circ}\)
8. \(x = 11\), \(m\angle TUH=20^{\circ}\)
9. No solution
10. \(x = 6.6\), \(m\angle LKV=16.8^{\circ}\)