Given:
A diagram of a circle O, [tex]arc(AB)=y^\circ, arc(CD)=(80+y)^\circ[/tex].
To find:
The measure of arc DC.
Solution:
According to the angle of intersecting chords theorem:
Angle at intersection of chords = Half of the sum of intercepted arcs.
Let [tex]\theta [/tex] be the angle between the intersection of chords AC and BD.
[tex]\theta +112^\circ=180^\circ[/tex] (Linear pair)
[tex]\theta =180^\circ-112^\circ[/tex]
[tex]\theta =68^\circ[/tex]
Using the angle of intersecting chords theorem, we get
[tex]\theta=\dfrac{arc(AB)+arc(CD)}{2}[/tex]
[tex]68^\circ =\dfrac{y^\circ+(80+y)^\circ}{2}[/tex]
[tex]136^\circ =(80+2y)^\circ[/tex]
On simplification, we get
[tex]136=80+2y[/tex]
[tex]136-80=2y[/tex]
[tex]56=2y[/tex]
Divide both sides by 2.
[tex]28=y[/tex]
Now,
[tex]arc(DC)=(80+y)^\circ[/tex]
[tex]arc(DC)=(80+28)^\circ[/tex]
[tex]arc(DC)=108^\circ[/tex]
Therefore, the correct option is C.
