Question

∆PQR and ∆SQR are isosceles triangles on the same base QR. Prove that <PQS = <PRS.

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Answer 1

Step-by-step explanation:

Given : Two isosceles triangles PQR and SQR having same base QR and SQ = SR and PQ = PR

To Prove : SP is perpendicular bisector of QR.

Proof : ∵∆PQR is an isosceles triangle. ∠PQ = PR Now ∠PQR = ∠PRQ In isosceles triangle, angles opposite to equal sides are same.

In ∆PQM and ∆PRM, PQ = PR (Given) ∠PQR = ∠PRQ PM = PM (Common)

∴ ∆PQM ≅ ∆PRM

∴ ∠QMP = ∠RMP and QM = MR Now, ∠QMP + ∠RMP = 180°

∠QMP = ∠RMP = 90° Thus. PM ⊥ QR ∴ M is mid-point of QR.

Similarly perpendicular bisector of QR is PM. Now point P lies on SM.

∵ ∆SQR is an isosceles triangle. Perpendicular bisector of QR is SM.

Therefore line segment SP will be perpendicular bisector of base