Question

Which of the following statements is True
A. All the angles of a parallelogram can be acute.
B. If the diagonals of a quadrilateral are at right angles it is a rhombus
C. If the diagonals of a quadrilateral are at right angles the figure formed by joining the mid points of the adjacent sides is a rectangle.
D. If one pair of opposite sides is parallel and the other pair of opposite sides is equal in a quadrilateral, it is a parallelogram.

Answer 1

Hint:
This is a statement based, Hence we will look into each statement and select or eliminate the ones which are true or false. In this question, We must know all the properties of quadrilaterals and their special types (square, rectangle, parallelogram, rhombus, etc). Hence, we will then find the true statement to get to the final answer.

Complete step by step solution:
According to the question
Statement-1:
 All the angles of a parallelogram can be acute.
We know that according to the angle sum property of a parallelogram (quadrilateral), The sum of all the interior angles of a parallelogram must be equal to 180. If we add four acute angles, the sum would always be less than 180. Hence, Statement-1 is false.
Statement-1 is eliminated.

Statement-2:
If the diagonals of a quadrilateral are at right angles it is a rhombus
We can simply eliminate this option if we can imagine a quadrilateral where the diagonals are perpendicular and all the four sides are not equal. This type of quadrilateral is also known as a kite. The figure of a kite is
According to the figure, though the diagonals are at right angles but all the four sides are not equal, Hence, it is not a rhombus.
Hence, Statement-1 is false.
Statement-1 is eliminated.

Statement-3:
If the diagonals of a quadrilateral are at right angles the figure formed by joining the mid points of the adjacent sides is a rectangle.
In Geometry, we have Three Quadrilaterals whose Diagonal are perpendicular to each other they are:-
Square
Rhombus
Kite
In Square, Rhombus, and Kite due to symmetry if the diagonals of these quadrilaterals are at right angles the figure formed by joining the midpoints of the adjacent sides is a rectangle
Hence this statement is true
Statement-3 is Selected.

Statement-4:
If one pair of opposite sides is parallel and the other pair of opposite sides is equal in a quadrilateral , it is a parallelogram.
The above statement can also be true for a trapezium which is not a parallelogram,
Hence this statement is False
Statement-4 is Eliminated.
Hence, only Statement-3 is correct,

Hence, the final answer is option C.

Note:
Sometimes, we are not able to confidently say that a particular choice is correct according to the theory we remember, in that case, we can read all the choices and eliminate those which are strongly false according to the theory we remember. This would narrow down the chances of getting our answer wrong and most of the time may result in getting a correct answer.