Question

In the given figure, if seg PR ≌ seg PQ, show that seg PS > seg

PQ.​

Answer 1

Given:

→seg PR≅seg PQ

To prove:

→seg PS > seg PQ

Solution:

In △ PQR,

seg PR≅seg PQ(given)

∴△PQR is an isoceles triangle.

→ If seg PR≅seg PQ

then,  ∠PQR≅∠PRQ( isoceles triangle theorem)---(1)

∴∠PRQ is an exterior angle of △PRS

→∠PRQ >∠PSR (Exterior angle property)----(2)

From (1) and (2)

∠PQR >∠PSR

i.e ∠Q >∠S

→PS > PQ(side opposite to greater angle is greater)

∴seg PS > seg PQ

Hence proved

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☆What is an isoceles triangle?

A triangle having two sides of same length is called an isoceles triangle △

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☆ What is isoceles triangle △ theorem?

According to this theorem,

(i) Angles opposite to equal sides are equal.

(ii) Sides opposite to equal angles are equal.

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☆ What is Exterior angle property?

An exterior angle of a triangle is equal to the sum of the opposite interior angles.

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Answer 2

In $\triangle PRS,\angle PRS$ is an obtuse angle.

Since, a triangle can have maximum 1 obtuse angle.

Hence,$\angle PRS$ is the greatest angle.

$\therefore$the side opposite to $\angle PRS$ i.e. PS is the longest side.

Now,Seg PS > Seg PR > Seg RS

$\therefore seg\:PS > seg\:PQ\\\:\:\:\: \:\:\because\:segPR≅ seg\:PQ$