# NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 4

Q-7 ABCD is a rhombus. Show that diagonal bisects as well as and

Diagonal BD bisects as well as.

Solution:

Given,

• ABCD is a rhombus

• AC and BD are its diagonals.

Proof,

• (Sides of a rhombus are equal)

• ………..equation (1) (Angles opposite of equal sides of a triangle are equal.)

• Also, , therefore, ……..equation (2) (Alternate interior angles)

From equation (1) and (2),

Therefore, AC bisects.

Similarly,

• (Sides of a rhombus are equal)

• …….equation (3) (Angles opposite of equal sides of a triangle are equal.)

Also, and so …..equation (4) (Alternate interior angles)

From Equation (3) and (4)

Therefore, AC bisects.

Similarly, diagonal BD bisects as well as.

Q-8 ABCD is a rectangle in which diagonal bisects as well as. Show that:

1. ABCD is a square

2. Diagonal BD bisects ∠B as well as ∠D.

Solution:

• Given ABCD is a rectangle, AC is bisects, as well as

Solution (i):

• (AC bisects as well as )

• (Sides opposite to equal angles of a triangle are equal)

• Also, (Opposite sides of a rectangle)

• Therefore,

• Thus, ABCD is a square.

Solution (ii):

• (Angles opposite to equal sides are equal)

• Also, (Alternate interior angles) therefore,

• Thus, BD bisects

Now,

Thus, BD bisects

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