Question

In the quadrilateral $PQRS$, the angles are in the ratio of $3:4:5:6$. Find the angles of $PQRS$. What is the special name given to $PQRS$?

Answer 1

Ratios of angles are $3:4:5:6$

Let the angles be $P=3x,Q=4x,R=5x$ and $S=6x$.

Sum of angles of the quadrilateral is ${360}^o.$

$\therefore$ $3x+4x+5x+6x={360}^o$

$\Rightarrow$ $18x={360}^o$

$\Rightarrow$ $x=\frac{{360}^o}{18}$

$\Rightarrow$ $x={20}^o$

$\Rightarrow$ $\angle P=3x=3\times {20}^o={60}^o$

$\Rightarrow$ $\angle Q=4x=4\times {20}^o={80}^o$

$\Rightarrow$ $\angle R=5x=5\times {20}^o={100}^o$

$\Rightarrow$ $\angle S=6x=6\times {20}^o={120}^o$

$\Rightarrow$ $\angle P+\angle S={60}^o+{120}^o={180}^o$

$\Rightarrow$ $\angle Q+\angle R={80}^o+{100}^o={180}^o$

$\therefore$ $PQ\parallel RS$

Since, one pair of quadrilateral is parallel then, it is trapezium.