Question

directions: use the diagram below. if ( mangle aec = 41^{circ} ), find each angle measure.

1. ( mangle aed ) 15. ( mangle cef )
2. ( mangle deb ) 17. ( mangle bef )
3. ( mangle ceb ) 19. ( mangle aef )
4. if ( angle pqr ) and ( angle sqt ) are vertical angles and ( mangle sqt = 109^{circ} ), find ( mangle pqr ).
5. if ( angle 2 ) and ( angle 3 ) are complementary angles and ( mangle 2 = 24^{circ} ), find ( mangle 3 ).
6. if ( angle k ) and ( angle l ) are supplementary angles and ( mangle k = 56^{circ} ), find ( mangle l ).

Explanation:

Step1: Vertical angles

Vertical angles are equal. For problem 20, since \( \angle PQR \) and \( \angle SQT \) are vertical angles, \( m\angle PQR=m\angle SQT \). Given \( m\angle SQT = 109^{\circ} \), so \( m\angle PQR=109^{\circ} \).

Step2: Complementary angles

Complementary angles sum to \( 90^{\circ} \). For problem 21, if \( \angle2 \) and \( \angle3 \) are complementary, then \( m\angle2 + m\angle3=90^{\circ} \). Given \( m\angle2 = 24^{\circ} \), we have \( m\angle3=90^{\circ}-24^{\circ}=66^{\circ} \).

Step3: Supplementary angles

Supplementary angles sum to \( 180^{\circ} \). For problem 22, if \( \angle K \) and \( \angle L \) are supplementary, then \( m\angle K + m\angle L=180^{\circ} \). Given \( m\angle K = 56^{\circ} \), we get \( m\angle L=180^{\circ}-56^{\circ}=124^{\circ} \).

Answer:

1. \( 109^{\circ} \)
2. \( 66^{\circ} \)
3. \( 124^{\circ} \)