Find in Fig. 5.26.

Question

Find in Fig. 5.26.

Homi · Accepted Answer

Understand the Question

A quadrilateral whose all four vertices lie on a circle is called a cyclic quadrilateral.
The sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$ (supplementary).
Here, all four vertices $A$ , $D$ , $C$ , and $B$ lie on the circle, making $ADCB$ a cyclic quadrilateral where $\angle ADC$ and $\angle ABC$ are opposite angles.

Step 1 · Find the Value of $x$

Since vertices $A$ , $D$ , $C$ , and $B$ lie on the circle, $ADCB$ is a cyclic quadrilateral. Diagram 1

The sum of opposite angles of a cyclic quadrilateral is $180^\circ$ $\angle ADC + \angle ABC = 180^\circ$

Substitute $\angle ADC = 100^\circ$ and $\angle ABC = x$

\begin{aligned} 100^\circ + x &= 180^\circ \\[0.6em] x &= 180^\circ - 100^\circ \\[0.6em] &= 80^\circ \end{aligned}

Answer

$x = 80^\circ$

Common Mistakes

Opposite vs. Adjacent Angles: Misidentifying which angles are opposite. In a cyclic quadrilateral, only opposite pairs sum to $180^\circ$ , not adjacent pairs.
Center Angle Confusion: Confusing the cyclic quadrilateral angle sum property ( $180^\circ$ ) with the theorem that an angle subtended at the center is double the angle at the circumference.

Question 3