Question

∆PQR is an isosceles triangle in which PQ = PR. Side QP is produced to M such that PM = PQ. Show that ∠QRM = 900

Answer 1

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Given:

△PQR is an isosceles triangle with PQ=PR

Proof:

PT bisects exterior angle ∠SPR and therefore ∠SPT=∠TPR=x⁰

∴ ∠Q=∠R (Property of an isosceles triangle)

also we know that in any triangle,

exterior angle= sum of the interior opposite angles.

∴ In △PQR, Exterior angle ∠SPR=∠PQR+∠PRQ

2x⁰ =∠Q+∠R

=∠Q+∠Q

2x⁰ =2∠Q

x⁰=∠Q

To prove:

PT∥QR

Lines PT and QR are cut by the transversal SQ. We have ∠SPT=x⁰

Hence, ∠SPT and ∠PQR are corresponding angles: PT∥QR

_____________________

Answer 2

Answer:

$\huge\sf \pmb{\orange {\underline \pink{\underline{\:Ꭺ ꪀ \mathfrak ꕶ᭙ꫀя \: }}}}$

Given:

△PQR is an isosceles triangle with PQ=PR

Proof:

PT bisects exterior angle ∠SPR and therefore ∠SPT=∠TPR=x⁰

∴ ∠Q=∠R (Property of an isosceles triangle)

also we know that in any triangle,

exterior angle= sum of the interior opposite angles.

∴ In △PQR, Exterior angle ∠SPR=∠PQR+∠PRQ

2x⁰ =∠Q+∠R

=∠Q+∠Q

2x⁰ =2∠Q

x⁰=∠Q

To prove:

PT∥QR

Lines PT and QR are cut by the transversal SQ. We have ∠SPT=x⁰

Hence, ∠SPT and ∠PQR are corresponding angles: PT∥QR

_____________________