Question

In triangle PQR, PQ = PR, QN = RM. Prove that
angle QPM = angle RPN.​

Answer 1

Given :- In triangle PQR, PQ = PR, QN = RM.

To Prove :- ∠QPM = ∠RPN .

Solution :-

In ∆PQR, we have,

→ PQ = PR .

so,

→ ∠PQR = ∠PRQ { Angle opposite to equal sides are equal .}

then,

→ ∠PQM = ∠PRN ------------ Eqn.(1)

now,

→ QN = RM (given)

→ QM + MN = RN + NM

→ QM = RN -------------- Eqn.(2)

now, in ∆PQM and ∆PRN we have,

→ PQ = PR { given }

→ ∠PQM = ∠PRN { from Eqn.(1) }

→ QM = RN { from Eqn.(2) }

therefore,

→ ∆PQM ≅ ∆PRN { By SAS congruence rule. }

hence,

→ ∠QPM = ∠RPN { By CPCT .} (Proved)

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