Question

1. if ( mangle abd = 79 ), what are ( mangle abc ) and ( mangle dbc )? (diagram: ( angle dbc = (5x + 4)^circ ), ( angle abc = (8x - 3)^circ )) 23. ( angle rqt ) is a straight angle. what are ( mangle rqs ) and ( mangle tqs )? (diagram: ( angle tqs = (6x + 20)^circ ), ( angle rqs = (2x + 4)^circ ))

Explanation:

Problem 22

Step1: Set up angle sum equation

We know that \( m\angle ABD = m\angle ABC + m\angle DBC \). From the diagram, \( m\angle DBC=(5x + 4)^\circ \) and \( m\angle ABC=(8x - 3)^\circ \), and \( m\angle ABD = 79^\circ \). So we set up the equation:
\( (5x + 4)+(8x - 3)=79 \)

Step2: Solve for x

Combine like terms:
\( 5x+8x + 4-3=79 \)
\( 13x + 1=79 \)
Subtract 1 from both sides:
\( 13x=79 - 1=78 \)
Divide both sides by 13:
\( x=\frac{78}{13}=6 \)

Step3: Find \( m\angle ABC \)

Substitute \( x = 6 \) into \( m\angle ABC=(8x - 3)^\circ \):
\( m\angle ABC=8\times6-3=48 - 3 = 45^\circ \)

Step4: Find \( m\angle DBC \)

Substitute \( x = 6 \) into \( m\angle DBC=(5x + 4)^\circ \):
\( m\angle DBC=5\times6+4=30 + 4 = 34^\circ \)

Step1: Set up angle sum equation (straight angle)

Since \( \angle RQT \) is a straight angle, \( m\angle RQT = 180^\circ \). And \( m\angle RQT=m\angle TQS + m\angle RQS \). From the diagram, \( m\angle TQS=(6x + 20)^\circ \) and \( m\angle RQS=(2x + 4)^\circ \). So we set up the equation:
\( (6x + 20)+(2x + 4)=180 \)

Step2: Solve for x

Combine like terms:
\( 6x+2x+20 + 4=180 \)
\( 8x+24 = 180 \)
Subtract 24 from both sides:
\( 8x=180 - 24=156 \)
Divide both sides by 8:
\( x=\frac{156}{8}=19.5 \)

Step3: Find \( m\angle RQS \)

Substitute \( x = 19.5 \) into \( m\angle RQS=(2x + 4)^\circ \):
\( m\angle RQS=2\times19.5+4=39 + 4 = 43^\circ \)

Step4: Find \( m\angle TQS \)

Substitute \( x = 19.5 \) into \( m\angle TQS=(6x + 20)^\circ \):
\( m\angle TQS=6\times19.5+20=117+20 = 137^\circ \)

Answer:

\( m\angle ABC = 45^\circ \), \( m\angle DBC = 34^\circ \)

Problem 23