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 ( overline{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: Use vertical angles property for \(x\)

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

Step2: Use linear - pair property for \(y\)

\(\angle DEH\) is a straight - angle (\(180^{\circ}\)). Since \(\angle DEG=(5x - 4)^{\circ}\) and \(x = 2\), \(\angle DEG=(5\times2 - 4)^{\circ}=6^{\circ}\).
\(\angle DEG\) and \(\angle DEH\) are supplementary (linear pair). So, \(6+(9y + 5)=180\).
Simplify: \(9y+11 = 180\).
Subtract \(11\): \(9y=169\).
Divide by \(9\): \(y = 19\).

Answer:

\(x = 2,y = 19\)