Question

The diagram shows a circle with center O. Chords AC and BC are drawn such that mAB and mCD are 42 degrees and 68 degrees, respectively. What is m angle DEC?

Answer 1

The measure of angle DEC is 1/2 times the sum of the 2 arcs. The arcs measure 110 degrees, so the angle DEC is half of that at 55 degrees.

Answer 2

Answer: $55^{\circ}$

Step-by-step explanation:  

Since, The angle subtended at the center of a circle is double the size of the angle subtended at the edge from the same two points,

Therefore,$\angle AOB=2\angle ADB$

$\implies\angle ADB=\frac{\angle AOB}{2}$

But, $\angle AOB=42^{\circ}$

Therefore, $\angle ADB=21^{\circ}$

Similarly, $\angle CAD=34^{\circ}$ ( because, it is given that,$\angle COD=68^{\circ}$)

Since, $\angle DEC$ is the exterior angle of $\triangle AED$

Thus, $\angle DEC=\angle CAD+\angle ADB=34^{\circ}+21^{\circ}=55^{\circ}$

⇒$\angle DEC=55^{\circ}$