# NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 4 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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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