Question

1. given ( mangle abc = 111^circ ). find ( mangle abd ) and ( mangle cbd ).

(diagram: rays ( ba ), ( bd ), ( bc ) with ( angle abd = (-10x + 58)^circ ), ( angle cbd = (6x + 41)^circ ))
( mangle abd = square^circ )
( mangle cbd = square^circ )

Explanation:

Step1: Set up the angle sum equation

Since \( \angle ABC = \angle ABD + \angle CBD \), we have \( (-10x + 58) + (6x + 41) = 111 \).
Simplify the left side: \( -10x + 58 + 6x + 41 = -4x + 99 \).
So the equation is \( -4x + 99 = 111 \).

Step2: Solve for x

Subtract 99 from both sides: \( -4x = 111 - 99 = 12 \).
Divide both sides by -4: \( x = \frac{12}{-4} = -3 \).

Step3: Find \( m\angle ABD \)

Substitute \( x = -3 \) into \( -10x + 58 \):
\( -10(-3) + 58 = 30 + 58 = 88 \).

Step4: Find \( m\angle CBD \)

Substitute \( x = -3 \) into \( 6x + 41 \):
\( 6(-3) + 41 = -18 + 41 = 23 \).

Answer:

\( m\angle ABD = \boxed{88}^\circ \)
\( m\angle CBD = \boxed{23}^\circ \)