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=(\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)`