AB is a diameter of a circle. C and D are points on the opposite sides of the diameter AB, such that $\angle ACD = 25^\circ$. E is a point on the minor are BD. Find the measure of $\angle BED$ (in degrees).

115

AB is the diameter of the circle.

\(\angle\)ACD = \({25}^\circ\)

\(\angle\)ACD = \(\angle\)ABD = \({25}^\circ\)  (Angles formed on the same arc re equal.)

\(\angle\)ADB = \({90}^\circ\)  (Angle formed by diameter in a semicircle)

In \(\Delta \)ADB

\(\angle\)BAD = \({180}^\circ\) - (\(\angle\)ADB + \(\angle\)ABD)

= \({180}^\circ\) - (\({90}^\circ\) + \({25}^\circ\)) = \({65}^\circ\)

Therefore, \(\angle\)BAD = \({65}^\circ\)

Now, In quadrilateral ABED,

\(\angle\)BAD + \(\angle\)BED = \({180}^\circ\)  (Sum of opposite angles are \({180}^\circ\))

= \({65}^\circ\) + \(\angle\)BED = \({180}^\circ\)

= \(\angle\)BED = \({180}^\circ\) - \({65}^\circ\) = \({115}^\circ\)

Therefore, \(\angle\)BED = \({115}^\circ\).