Question

30 POINTS!!

The circle below has center C. Suppose angle E D F equals 81 degree and stack D F with left right arrow on top is tangent to the circle at D.


Find the following. Type your numerical answers (without units) in each blank.

Answer 1

If we assume that "mED" means the measure of angle EDC (since D is a point on the circle and E and F are not specified), and "M<ECD" means the measure of angle ECD, then we can use the fact that an angle inscribed in a circle is half the measure of the intercepted arc to solve the problem.

Since DF is tangent to the circle at D, we know that angle EDF is a right angle (90 degrees). Therefore, angle EDC is the sum of angles EDF and FDC, which is 90 degrees plus the measure of angle FDC. Since DF is tangent to the circle, angle FDC is equal to the measure of the intercepted arc FD, which is also 81 degrees. Therefore, angle EDC is 90 + 81 = 171 degrees.

Since angle ECD is an inscribed angle that intercepts the same arc as angle FDC, it must also have a measure of 81 degrees. Therefore, M<ECD = 81 degrees.

Answer 2

162 degrees

Explanation:

If we let G be a point on the circumference in the major segment of the circle (to the right of ED), then due to the alternate segment theorem, angle EGD = angle EDF = 81 degrees.

Then to get angle ECD we multiply by 2, since the angle at the centre is twice the angle at the circumference.