ABCD is a cyclic quadrilateral with angle ABC = 136 degrees. Find angle ADC.

Step-by-step solution

1. 1

   Identify which pair of angles the theorem applies to. $\angle ABC$ is the interior angle at $B$ and $\angle ADC$ is the interior angle at $D$. In the quadrilateral $ABCD$ the vertices $B$ and $D$ are opposite, which is exactly the pair the cyclic-quadrilateral theorem constrains.

2. 2

   State the theorem. In any quadrilateral inscribed in a circle, opposite angles are supplementary:

   $\angle ABC+\angle ADC=180^{\circ}$

   (The reason: the two angles are inscribed angles subtending the two arcs $ADC$ and $ABC$, which together make the whole $360^{\circ}$ circle, and each inscribed angle is half its arc.)

3. 3

   Substitute the given value.

   $136^{\circ}+\angle ADC=180^{\circ}$

4. 4

   Solve.

   $\angle ADC=180^{\circ}-136^{\circ}=44^{\circ}$

   $\boxed{44^{\circ}}$

5. 5

   Sanity-check the configuration. $\angle ABC$ is obtuse, so its opposite angle must be acute — and $44^{\circ}$ is. The other pair, $\angle BAD$ and $\angle BCD$, must also sum to $180^{\circ}$, so all four interior angles total $360^{\circ}$ as required for any quadrilateral. Note the theorem gives no information about the individual values of $\angle BAD$ and $\angle BCD$ from this data alone.