Question

In an isosceles trapezium, $\angle C$ is equal to
A. ${115^ \circ }$
B. ${65^ \circ }$
C. ${105^ \circ }$
D. ${75^ \circ }$

Answer 1

Hint: The first hint to do this question is that it is an isosceles trapezium. One pair of opposite sides in an isosceles trapezium are parallel. There is one property of an isosceles triangle related to angles like angles adjacent to non-parallel sides are equal. Also, an important property of quadrilaterals is used here, that is angle sum property.

Complete step by step answer:
In the given question, we know that it is an isosceles trapezium. So, the sides AB and CD are parallel. Therefore,
$\angle A = \angle B$
From the given figure, we know that $\angle A = {115^ \circ }$. So,
$\angle B = {115^ \circ }$
We also know that
$\angle C = \angle D$
Now, using angle sum property
$\angle A + \angle B + \angle C + \angle D = {360^ \circ }$

Now putting the values of $\angle A\,\,and\,\,\angle B$
We know that $\angle C = \angle D$
${115^ \circ } + {115^ \circ } + \angle C + \angle C = {360^ \circ }$
On simplification, we get
${230^ \circ } + 2\angle C = {360^ \circ }$
On transposing, we get
$2\angle C = {360^ \circ } - {230^ \circ }$
$\Rightarrow 2\angle C = {130^ \circ }$
On dividing, we get
$\angle C = {65^ \circ }$
Therefore, the value of $\angle C$is ${65^ \circ }$.

Hence, the correct option is $\left( B \right)$.

Additional information: There are some Properties of an Isosceles Trapezium. Only one pair of sides are parallel. Non-parallel sides (legs) are equal in measure. The diagonals are equal in measure. The base angles are equal in measure. The opposite angles are supplementary. The segment which joins the midpoints of the parallel sides is perpendicular to them. An isosceles trapezium can be inscribed in a circle.

Note: A trapezium is a quadrilateral in which only one pair of opposite sides are parallel to each other. An isosceles trapezium is a trapezium in which the non-parallel sides are equal in measure. In other words, the bases are parallel and the legs are equal in measure. In this article, we will learn the properties of an isosceles trapezium.