# In the given figure, ABC is an equilateral triangle, PQ ‖AC and AC is produced to R such that CR=BP. Prove that QR bisects PC.

Given: ABC is an equilateral triangle, PQ AC and CR=BP

To prove: QR bisects PC or PM = MC

Proof:

Since, ∆ABC is equilateral triangle,

A = ACB = 60°

Since, PQ AC and corresponding angles are equal,

BPQ = ACB = 60°

In ∆BPQ,

B= ACB = 60°

BPQ = 60°

Hence, ∆BPQ is an equilateral triangle.

PQ = BP = BQ

Since we have BP = CR,

We say that PQ = CR …(1)

Consider the triangles ∆PMQ and ∆CMR,

PQM = CRM …alternate angles

PMQ = CMR … vertically opposite angles

PQ = CR … from 1

Thus by AAS property of congruence,

∆PMQ CMR

Hence, we know that, corresponding parts of the congruent triangles are equal

PM = MC

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