To find the measure of angle \( \angle EGC \), we can use the property of angles formed by intersecting chords in a circle. The measure of the angle formed by two intersecting chords is equal to half the sum of the measures of the arcs intercepted by the angle.
In this case, we have:
- Arc \( EC \) measuring \( 50^\circ \)
- Arc \( DF \) measuring \( 10x \) degrees
According to the angle construction, we have: \[ \angle EGC = \frac{1}{2} \left( \text{measure of arc EC} + \text{measure of arc DF} \right) \]
Thus, we can write the equation: \[ \angle EGC = \frac{1}{2} \left( 50 + 10x \right) \]
Since we are also given that \( \angle EGC = 7x + 7 \), we can set the two expressions for \( \angle EGC \) equal to each other: \[ 7x + 7 = \frac{1}{2}(50 + 10x) \]
Now, let's solve for \( x \):
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First, multiply both sides by 2 to eliminate the fraction: \[ 2(7x + 7) = 50 + 10x \] \[ 14x + 14 = 50 + 10x \]
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Rearranging gives: \[ 14x - 10x = 50 - 14 \] \[ 4x = 36 \] \[ x = 9 \]
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Now substitute \( x = 9 \) back into the formula for \( \angle EGC \): \[ \angle EGC = 7(9) + 7 = 63 + 7 = 70^\circ \]
The measure of \( \angle EGC \) is \( \boxed{70^\circ} \).
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