# NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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.