Question

ABCD is a rhombus with diagonals AC and BD.
$\text{The value of}p+q+r+s+t\text{is}$ .

• A. ${270}^{\circ }$
• B. ${180}^{\circ }$
• C. ${540}^{\circ }$
• D. ${360}^{\circ }$

Answer 1

The correct option is A ${270}^{\circ }$

In the rhombus ABCD,
$\angle A+\angle B+\angle C+\angle D={360}^{\circ }$ [Angle sum property]

$\text{In}△AED\text{and}△AEB,AB=AD\text{[Sides of rhombus]}AE=AE\text{[Common side]}\angle AED=\angle AEB\text{[Diagonals of a rhombus bisect at}{90}^{\circ }]\therefore △AED\cong △AEB$

$\therefore \angle DAC=\angle CAB=\frac{1}{2}\angle DAB$
$\text{[CPCT]}$
$\Rightarrow p=\frac{1}{2}\angle A$
$\Rightarrow \angle A=2p$

Similarly,
$\angle B=2q$, $\angle C=2t$ and $\angle D=2s$
$\Rightarrow 2p+2q+2t+2s={360}^{\circ }$
$\Rightarrow p+q+t+s={180}^{\circ }$

The diagonals of a rhombus intersect at ${90}^{\circ }$
$\therefore \angle AEB=r={90}^{\circ }$

Therefore, $p+q+r+s+t={180}^{\circ }+{90}^{\circ }$
$\Rightarrow p+q+r+s+t={270}^{\circ }$