Question

Therom 8.10 in quadrilateral chapter convers statmant of therom 8.9​

Answer 1

Answer:

THEOREM : Diagonals of a rhombus are perpendicular to each other.

Step-by-step explanation:

As given, let PQRS is a rhombus and PR and SQ are its diagonals.

Then to prove diagonals PR and SQ are perpendicular to each other.

That is ∠POQ = 90o

Proof :

We know that all sides are equal in a rhombus.

Thus, in rhombus PQRS,

PQ = QR = RS = PS

Again we know that a rhombus is a parallelogram, and diagonals of a parallelogram bisects each other.

Thus, in the given rhombus

OP = OR

Now, in ΔPOQ and ΔQOR,

OP = OR

And PQ = QR [Sides of rhombus]

And OQ is common side

Thus, from SSS (Side Side Side) congruency

ΔPOQ ≅ ΔQOR

Now from CPCT we know that corresponding parts of congruent triangles are equal.

Thus, ∠POQ = ∠QOR

And, since ∠POQ and ∠QOR together form linear pair of angles

Thus, ∠POQ + ∠QOR = 180o

⇒ ∠POQ + ∠POQ = 180o

[Since, ∠POQ = ∠QOR]

⇒ 2 ∠POQ = 180o

⇒ ∠POQ = 180o/2

⇒ ∠ = POQ = 90o

Thus, ∠QOR = 90o

Now, we know that vertically opposite angles are equal

Thus, ∠POQ = ∠ROS = ∠QOR = ∠POS = 90o

Thus, diagonals PR and SQ are perpendicular to each other Proved

Thus, diagonals of a rhombus are perpendicular to each other. Proved