Question

ABCD is a trapezium such that AB||CD, $\angle A:\angle D = 2:1$, $\angle B:\angle C = 7:5$. Find the angles of the trapezium.

Answer 1

Hint: ABCD is a trapezium. A trapezium is a quadrilateral in which any two sides are parallel. The ratio of angles is given in the question. Sum of all the angles of a quadrilateral should be ${360^ \circ }$. We can proceed by assuming $\angle A = 2x$, $\angle D = x$ and $\angle B = 7y,\angle C = 5y$ because the ratios are already given. one pair of sides is parallel in case of trapezium. So, the sum of co interior angles will be ${180^ \circ }$(parallel line property).

Complete Step by Step Solution:
The quadrilateral ABCD given below is a trapezium in which AB||CD.
As, $\angle A:\angle D = 2:1$ and $\angle B:\angle C = 7:5$
Let $\angle A = 2x$, $\angle D = x$ and $\angle B = 7y,\angle C = 5y$

$\because $AB||CD $ \Rightarrow \angle A + \angle D = {180^ \circ }$( Sum of co interior angles between a parallel line is always ${180^ \circ }$)
$ \Rightarrow 2x + x = {180^ \circ }$
$ \Rightarrow 3x = {180^ \circ }$
$ \Rightarrow x = \dfrac{{{{180}^ \circ }}}{3} = {60^ \circ }$
$
  \therefore \angle A = 2x = 2 \times {60^ \circ } = {120^ \circ } \\
  \angle D = x = {60^ \circ } \\
 $
Also, $\angle B + \angle C = {180^ \circ }$( Sum of co interior angles between a parallel line is always ${180^ \circ }$)
$ \Rightarrow 7y + 5y = {180^ \circ }$
$ \Rightarrow 12y = {180^ \circ }$
$ \Rightarrow y = \dfrac{{{{180}^ \circ }}}{{12}} = {15^ \circ }$
$
  \therefore \angle B = 7y = 7 \times {15^ \circ } = {105^ \circ } \\
  \angle C = 5y = 5 \times {15^ \circ } = {75^ \circ } \\
 $
So, $\angle A = {120^ \circ },\angle B = {105^ \circ },\angle C = {75^ \circ }$and $\angle D = {60^ \circ }$
We can also cross verify our solution.
As, the sum of all the angles of a quadrilateral is ${360^ \circ }$.
$\therefore \angle A + \angle B + \angle C + \angle D$Should be ${360^ \circ }$.
$\angle A + \angle B + \angle C + \angle D = {180^ \circ }$
LHS= ${120^ \circ } + {105^ \circ } + {75^ \circ } + {60^ \circ } = {360^ \circ }$
RHS= ${360^ \circ }$
So, LHS= RHS
Therefore, our solution is correct.

Note:
Students generally use the wrong property that the co interior angles are equal but they are not equal. Their sum is ${180^ \circ }$. Take care of the properties and do calculations properly.