Question

in figure 10.37, given below,∆P and ∆PRN are equilateral triangles. Prove that i) angle MPR = angle QPN, ii) MR = QN
Please help?(a/n: an important thing is that it is not given that the equilateral triangles are congruent to each other.)
I had asked this question earlier but forgot to attach the diagram. Silly me!

Answer 1

In order to prove this, the two equilateral triangles must be congruent. The data provided isn't enough perhaps.

Answer 2

In ΔMQP

In ΔMQP∠MQP=60°,∠QMP=60°,∠MPQ=60°(All angles of an equilateral triangle are equal i.e=60°)

In ΔPNR

In ΔPNR ∠NPR=60°,∠PNR=60°,∠PRN=60°(Equilateral triangle)

In ΔMPQ and ΔPQR

PQ=PQ(Common Side)

∠QPM=∠PQR(Alternate interior opposite angles) or ∠PQR=60°

[[As ∠NPR=60° and ∠NPR=∠PRQ(Alternate interior angles)

So ∠PRQ=60°]]

∠PMQ=∠PRQ(60°each)

ΔMQP≅ΔPQR(AAS congruency)

In ΔPNR and ΔPQR

PR=PR(Common)

∠PRQ=∠RPN(Alternate interior angle)

∠PNR=∠PQR(60° each)

ΔPNR≅ΔPQR(AAS congruency)

So ΔPNR≅ΔMQP

∠MPQ=∠NPR (cpct)

In quadrilateral MPQR and PQRN

∠MPQ+∠QPR=∠MPR

and ∠NPR+∠QPR=∠QPN

So ∠MPR=∠QPN proved

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