Question

Is it possible to construct a unique quadrilateral ABCD with AB = 4cm, AD = 6 cm, $\angle ABC={60}^{\circ }$, $\angle DAB={75}^{\circ }$, and $\angle ADC={85}^{\circ }$? If yes, find the value of $\angle BCD$.

• A. No, it is impossible
• B. Yes, ${130}^{\circ }$
• C. Yes, ${140}^{\circ }$
• D. Yes, ${150}^{\circ }$

Answer 1

The correct option is C

Yes, ${140}^{\circ }$

We have 5 measurements. Thus, a quadrilateral can be constructed.

Step1: Draw AB = 4 cm.

Step 2: Draw $\angle DAB={75}^{\circ }$ and mark AD = 6 cm on it.

Step 3: At D, draw $\angle ADC={85}^{\circ }$ and extend the line along $\angle ADC$.

Step 4: Draw $\angle ABC={60}^{\circ }$ and extend the line along $\angle ABC$ such that it intersects the line extending along $\angle ADC$. Mark the point of intersection as C.

Step 5: Join the points A,B,C and D. Measure $\angle BCD$. It is equal to ${140}^{\circ }$ _.

Alternatively :

$\angle BCD$ $={360}^{\circ }$ - $(\angle ABC+\angle DAB+\angle ADC)$ = ${360}^{\circ }$ - $({60}^{\circ }+{75}^{\circ }+{85}^{\circ })$ = $({360}^{\circ }-{220}^{\circ })$ = ${140}^{\circ }$