NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 2
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 updown.
Q3 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 SideAngleSide 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.
Q4 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 SideAngleSide congruence condition.

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

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

Therefore by AngleAngleSide 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)