Question

If the angels of a quadrilateral, taken in order, are in the ratio $1:2:3:4$, prove that it is a trapezium.

Answer 1

Let the four angles be $x,2x,3x$ and $4x$

We know that, sum of four angles of a quadrilateral is ${360}^o.$

$\therefore$ $x+2x+3x+4x={360}^o$

$\Rightarrow$ $10x={360}^o$

$\Rightarrow$ $x={36}^o$

$\Rightarrow$ $2x=2\times {36}^o={72}^o$

$\Rightarrow$ $3x=3\times {36}^o={108}^o.$

$\Rightarrow$ $4x=4\times {36}^o={144}^o$

$\Rightarrow$ Measure of angles of four angles of quadrilateral are ${36}^o,{72}^o,{108}^o$ and ${144}^o.$

Since, measure of all angles are different then, it is trapezium.