Question

A quadrilateral is formed by intersecting two circles, such that two vertices of the quadrilateral lie on each circle. If one angle of the quadrilateral is right angle, comment on the properties of the quadrilateral.

• A. Two sides are parallel to each other.
• B. The quadrilateral has two right angles.
• C. The sum of opposite angles is ${180}^o.$
• D. The sum of adjacent angles is ${180}^o.$

Answer 1

Answer: (A), (B), (D)

The sum of adjacent angles is ${180}^o.$

The schematic of the quadrilateral $VWXY$ with the given conditions is as follows:
Connect the intersection points A and B.
Given: $\angle W={90}^o$

$XABW$ is a cyclic quadrilateral to the smaller circle.
As opposite angles of a cyclic quadrilateral have a sum of ${180}^o,$
$\angle XAB={180}^o-\angle W={90}^o$

XAY is a straight line.
$\therefore \angle YAB={180}^o-\angle XAB={90}^o$

AYVB is a cyclic quadrilateral to the larger circle.
$\angle V={180}^o-\angle YAB={90}^o$

$\therefore$ The quadrilateral has two right angles.
$\Rightarrow$ Option (b) is correct.

From the figure, as $\angle W$ and $\angle V$ are right angles, XW and YV must be parallel to each other.
$\Rightarrow$ Option (a) is correct.

Also, $\angle W$ and $\angle V$ are adjacent angles.
$\angle W+\angle V={180}^o$
$\Rightarrow$ Option (d) is correct.

$VWXY$ is not a cyclic quadrilateral.
$\therefore$ The sum of opposite angles may not be ${180}^o.$