Question

In the adjoining figure, PR =PT; ∠QPS =∠TPR and PQ = PS. Show that QR = ST.

Answer 1

Step-by-step explanation:

In the given figure, we are provided that,

PS = PT

PQ = PR

also,

∠QPR = ∠SPT

As we know that Angle opposite to the equal sides are also equal.

Therefore,

∠PQR = ∠PRQ = θ (say) (because, PQ = PR)

also,

∠PST = ∠PTS = ∅ (say) (because, PS = PT)

Now,

As,

Sum of angles in a triangle is equal to 180°.

Therefore, in triangle PQR,

∠QPR = 180° - ∠PQR - ∠PRQ = 180° - θ - θ = 180° - 20

Similarly, in triangle PST,

∠TPS = 180° - ∠PST - ∠PTS = 180° - ∅ - ∅ = 180° - 2∅

But,

∠QPR = ∠TPS  (given)

So,

180° - 2θ = 180° - 2∅

θ = ∅

Now,

In triangle PQS and triangle PRT,

PQ = PR (given)

∠QPS = 180° - 2θ + ∠RPS = ∠TPR = 180° - 2∅ + ∠RPS (because, θ = ∅)

and,

PS = PT (given)

Therefore,

Triangle PQS ≅ Triangle PRT (SAS congruency)

So,

QS = RT

Hence, proved.

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