# NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 3

Get unlimited access to the best preparation resource for NSTSE Class-9: fully solved questions with step-by-step explanation- practice your way to success.

Download PDF of This Page (Size: 144K) ↧

Co-interior Angles

When two lines are cut by a third line (transversal) co-interior angles are between the pair of lines on the same side of the transversal. If the lines are parallel the co-interior angles are supplementary (add up to 180 degrees).

Q-5 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 Side-Angle-Side 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 Side-Angle-Side congruence condition

Thus, by Corresponding Parts of Congruent Triangle………equation(2)

Also,

and

(From equation (1))

by Corresponding Parts of Congruent Triangle

(co-interior 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.

Q-6 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.