Question

1. if ( moverarc{cb} = (5x + 41)^circ ), ( moverarc{ad} = (7x - 19)^circ ), and ( mangle aed = 89^circ ), find ( moverarc{ad} ).

Explanation:

Step1: Recall intersecting chords angle theorem

The measure of an angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs. So:
$$m\angle AED = \frac{1}{2}(m\widehat{CB} + m\widehat{AD})$$

Step2: Substitute given values

Plug in $m\angle AED=89^\circ$, $m\widehat{CB}=(5x+41)^\circ$, $m\widehat{AD}=(7x-19)^\circ$:
$$89 = \frac{1}{2}[(5x + 41) + (7x - 19)]$$

Step3: Simplify the equation

First combine like terms inside the parentheses:
$$89 = \frac{1}{2}(12x + 22)$$
Multiply both sides by 2:
$$178 = 12x + 22$$

Step4: Solve for $x$

Subtract 22 from both sides:
$$178 - 22 = 12x$$
$$156 = 12x$$
Divide by 12:
$$x = \frac{156}{12} = 13$$

Step5: Calculate $m\widehat{AD}$

Substitute $x=13$ into $m\widehat{AD}=(7x-19)^\circ$:
$$m\widehat{AD} = 7(13) - 19$$
$$m\widehat{AD} = 91 - 19$$

Answer:

$72^\circ$