# NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Vertically opposite angles

Vertically Opposite Angles are the angles opposite each other when two lines cross. “Vertical” in this case means they share the same Vertex (or corner point) , not the usual meaning of up-down.

Q-3 Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Let ABCD be a quadrilateral whose diagonals bisect each other at right angles. Given,

To show,

ABCD is parallelogram and

Proof,

In and ,

(Given)

(Each , Opposite sides of a parallelogram are equal)

OB = OB (Common)

Therefore, by Side-Angle-Side congruence condition.

Thus, by Corresponding Parts of Congruent Triangles

Similarly, we can prove,

Opposites sides of a quadrilateral are equal hence ABCD is a parallelogram. Thus, ABCD is rhombus as it is a parallelogram whose diagonals intersect at right angle.

Q-4 Show that the diagonals of a square are equal and bisect each other at right angles.

Solution:

Let ABCD be a square and its diagonals AC and BD intersect each other at O. To show,

,

Proof,

In , (Common) (Given)

Therefore, by Side-Angle-Side congruence condition.

Thus, by Corresponding Parts of Congruent Triangles. Therefore, diagonals are equal.

Now, In (Alternate interior angles) (Vertically opposite triangle) (Given)

Therefore by Angle-Angle-Side congruence condition.

Thus, by Corresponding Parts of Congruent Triangles (Diagonal bisect each other.)

Now, , (Given) (diagonals are bisected) (Sides of the square)

Therefore, by SSS congruence condition. also, (Linear pair)

Thus, (Diagonals bisect each other at right angles)