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Answer :
Final answer:
To find the measure of mArc GEB in circle C, we use the measures of the intercepted arcs and central angles. By combining mArc GF (51 degrees), mArc FE (90 degrees), and mArc EB (39 degrees), we conclude that mArc GEB measures 180 degrees.
Explanation:
The question asks for the measure of mArc GEB in circle C. We're given three angles: Angle GCF is 51 degrees, angle FCE is 90 degrees, and angle DCE is 39 degrees. To solve for mArc GEB, we need to consider the entire circumference of the circle which is 360 degrees.
Since angle FCE is 90 degrees and it intercepts arc FE, arc FE is also 90 degrees in measure. Similarly, angle DCE intercepts arc DE, and since DCE is 39 degrees, arc DE is also 39 degrees in measure. Now, we can find the measure of mArc GD (which is the diameter, hence a straight line across the circle, and thus 180 degrees).
To find mArc GEB, we subtract arcs FE and DE from mArc GD. This calculation gives us:
mArc GEB = mArc GD - mArc FE - mArc DE
= 180° - 90° - 39°
= 51°
However, to find mArc GEB we actually need to add mArc GF, mArc FE, and mArc EB. mArc GF is the same as angle GCF, which is 51 degrees, as these are central angles.
Thus, mArc GEB = mArc GF + mArc FE + mArc EB
= 51° + 90° + 39°
= 180°.
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