Question

The measure of the angles of a quadrilateral $ABCD$ are respectively in the ratio $1:2:3:4$. Then which of the following is true

• A. $AC=BD$
• B. $AD\parallel BC$
• C. $AB\parallel DC$
• D. $noneofthese$

Answer 1

The correct option is D $noneofthese$

$\text{Let}\angle A=x,\Rightarrow B=2x,C=3x,D=4x$

We know that, sum of angles of quadrilateral $={360}^o$

$\Rightarrow x+2x+3x+4x={360}^o\Rightarrow =10x={360}^0,x={36}^o\text{Hence,}A.={36}^o,B={72}^o,C={108}^o\&D={144}^o$

Here, opposite angles are supplementary i.e. $A+C=B+D={180}^o$

Hence, it is a cyclic quadrilateral, But, in a cyclic quadrilateral , the diagonals may or may not pass through centre of circle. Hence, AC can or cannot be equal to BD.

Therefore, option D is correct.