Question

Construct a quadrilateral $PQRS$ in which $PQ=3.8$ cm, $QR=6.8$ cm and three angles $\angle P={100}^o,\angle R={110}^o$ and $\angle S={75}^o$ are known.

Answer 1

We know that the sum of all four angles of a quadrilateral is ${360}^{\circ }$

$\Rightarrow$ $\angle P+\angle Q+\angle R+\angle S={360}^{\circ }$

$\Rightarrow$ ${100}^{\circ }+\angle Q+\angle {110}^{\circ }+{75}^{\circ }={360}^{\circ }$

$\Rightarrow$ ${285}^{\circ }+\angle Q={360}^{\circ }$

$\therefore$ $\angle Q={75}^{\circ }$

Steps to construct a quadrilateral when two adjacent sides and three angles are known:
Step 1: Draw a line segment $PQ=3.8cm$

Step 2: Take $Q$ as the center and draw an angle of ${75}^{\circ }$

Step 3: Cut off $QR=6.8cm$

Step 4: Take $R$ as the center and draw an angle of ${110}^o$

Step 5: Take $P$ as the center and draw an angle of ${100}^{\circ },$ which cuts of $\angle R$ at point $S$.

$\therefore$ $PQRS$ is required quadrilateral.