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

Get unlimited access to the best preparation resource for CBSE/Class-9 : get questions, notes, tests, video lectures and more- for all subjects of CBSE/Class-9.

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

1. it bisects also,
2. 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.

Developed by: