Question

1. if ( mangle deg=(5x - 4)^{circ},mangle gef=(7x - 8)^{circ},mangle deh=(9y + 5)^{circ} ), find the values of ( x ) and ( y ).
2. ( angle r ) and ( angle s ) are complementary angles. if ( mangle r=(12x - 3)^{circ} ) and ( mangle s=(7x - 2)^{circ} ), find ( mangle r ).
3. ( angle p ) and ( angle q ) are supplementary angles. if ( mangle p=(4x + 1)^{circ} ) and ( mangle q=(9x - 3)^{circ} ), find ( mangle q ).
4. ( angle 1 ) and ( angle 2 ) form a linear pair. the measure of ( angle 2 ) is six more than twice the measure of ( angle 1 ). find ( mangle 2 ).
5. ( angle j ) and ( angle k ) are complementary angles. the measure of ( angle j ) is 18 less than the measure of ( angle k ). find the measure of each angle.
6. if ( overrightarrow{uw} ) bisects ( angle tuv,mangle tuw=(13x - 5)^{circ} ) and ( mangle wuv=(7x + 31)^{circ} ), find the value of ( x ).
7. if ( overrightarrow{mo} ) bisects ( angle pmn,mangle pmn = 74^{circ} ) and ( mangle omn=(2x + 7)^{circ} ), find the value of ( x ).
8. 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: Find \(x\) using vertical angles

Since \(\angle DEG\) and \(\angle GEF\) are vertical angles, \(m\angle DEG=m\angle GEF\).
So, \(5x - 4=7x - 8\).
Subtract \(5x\) from both sides: \(-4 = 2x-8\).
Add \(8\) to both sides: \(4 = 2x\).
Divide both sides by \(2\): \(x = 2\).

Step2: Find \(y\) using linear - pair angles

\(\angle DEH\) is a straight - angle (\(180^{\circ}\)).
Since \(x = 2\), \(m\angle DEG=5x - 4=5\times2 - 4=6^{\circ}\), \(m\angle GEF=7x - 8=7\times2 - 8 = 6^{\circ}\).
\(\angle DEH\) is a straight - angle. \(\angle DEG\) and \(\angle GEF\) are vertical angles. \(\angle DEH\) is a linear - pair with the sum of \(\angle DEG\) and \(\angle GEF\) (or \(2\angle DEG\) since \(\angle DEG=\angle GEF\)).
\(m\angle DEH=(9y + 5)^{\circ}\) and \(m\angle DEH = 180^{\circ}\).
So, \(9y+5 = 180\).
Subtract \(5\) from both sides: \(9y=175\).
Divide both sides by \(9\): \(y=\frac{175}{9}\approx19.44\).

Answer:

\(x = 2\), \(y=\frac{175}{9}\)