NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 3
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Cointerior Angles
When two lines are cut by a third line (transversal) cointerior angles are between the pair of lines on the same side of the transversal. If the lines are parallel the cointerior angles are supplementary (add up to 180 degrees).
Q5 Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
Solution:

Given, ABCD is a quadrilateral in which diagonals AC and BD bisect each other at right angle at O.

To prove, Quadrilateral ABCD is a square.
Proof,
In ,

(Diagonals bisect each other)

(Vertically opposite)

(Diagonals bisect each other)
Therefore,

by SideAngleSide congruence condition.

Thus, by Corresponding Parts of Congruent Triangle…….equation(1)
Also,

(Alternate interior angles)

Now, In ,

(Diagonals bisect each other)

(Vertically opposite)

(Common)
Therefore, by SideAngleSide congruence condition
Thus, by Corresponding Parts of Congruent Triangle………equation(2)
Also,

and

(From equation (1))

by Corresponding Parts of Congruent Triangle

(cointerior angles)

( )

……..equation(3)

That is, one of the interior angle is right angle.
Thus, from (1), (2) and (3) given quadrilateral ABCD is a square.
Q6 Diagonal AC of a parallelogram ABCD bisects (see Fig.). Show that

it bisects also,

ABCD is a rhombus.
Solution:
Given, Parallelogram ABCD and its diagonal AC, it’s bisects
In ,

(parallel sides are equal in a parallelogram)

(parallel sides are equal in a parallelogram)

(Common side)
Therefore, by SSS congruence condition.
Thus,

by Corresponding Parts of Congruent Triangle

And (AC id diagonal)

(AC id diagonal)
Therefore, AC bisects also.
Since, (Proved)

(Opposite sides of equal angles of a triangle are equal)

Also, (parallel sides are equal in a parallelogram)
Thus, ABCD is a rhombus.