We are asked to find the measures of arc $BDC$ and arc $BC$ in the given circle diagram. Angle $ABC$ is an exterior angle measuring $124^\circ$.

1. Problem Description

We are asked to find the measures of arc $BDC$ and arc $BC$ in the given circle diagram. Angle $ABC$ is an exterior angle measuring $124^\circ$.

2. Solution Steps

Since angle $ABC$ is an exterior angle, we know that the angle formed by the intersecting lines is half the difference of the intercepted arcs. Let $x$ represent the measure of arc $BC$. Then the measure of arc $BDC$ is $360 - x$. Therefore,

$\frac{1}{2}|(360-x) - x| = 124$

$|360 - 2x| = 248$

We have two cases:

Case 1: $360 - 2x = 248$

$2x = 360 - 248 = 112$

$x = 56$

Case 2: $360 - 2x = -248$

$2x = 360 + 248 = 608$

$x = 304$

Since arc $BC$ is intercepted by the exterior angle, it must be smaller than arc $BDC$, thus $m\stackrel{\frown}{BC}$ must be less than

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8

0. We choose $x=56$ as the measure of arc $BC$ because if we choose $x=304$, we are looking at the reflex angle.

Therefore, the measure of arc $BC$ is $56^\circ$.

Now we can find the measure of arc $BDC$.

$m\stackrel{\frown}{BDC} = 360 - m\stackrel{\frown}{BC} = 360 - 56 = 304$.

3. Final Answer

The mBDC is $304^\circ$ and the mBC is $56^\circ$.