Question

$\huge \fbox \red{❥ Question}$

In the given figure, PS/SQ = PT/TR and ∠ PST = ∠ PRQ. Prove that PQR is an isosceles triangle.
✳️ give proper answer brainlian..​

Answer 1

Question:

In the given figure, PS/SQ = PT/TR and ∠ PST = ∠ PRQ. Prove that PQR is an isosceles triangle.

Answer:

We have,

SQ

PS = TR

PT⇒ ST∣∣QR [By using the converse of Basic Proportionality Theorem]

⇒ ∠PST=∠PQR [Corresponding angles]

⇒ ∠PRQ=∠PQR [∵∠PST=∠PRQ (Given)]

⇒ PQ=PR [∵ Sides opposite to equal angles are equal]

⇒ △PQR is isosceles.ANSWER

We have,

SQPS

= TRPP

⇒ ST∣∣QR [By using the converse of Basic Proportionality Theorem]

⇒ ∠PST=∠PQR [Corresponding angles]

⇒ ∠PRQ=∠PQR [∵∠PST=∠PRQ (Given)]

⇒ PQ=PR [∵ Sides opposite to equal angles are equal]

⇒ △PQR is isosceles..

Not Sure About this answer...

Answer 2

Given :-

$\bf \frac{PS} {SQ} = \frac{PT} {TR}$

$\bf∠ \: PST = ∠ \: PRQ$

To prove :-

PQR is an isosceles triangles

Proof :-

$\bf Given \: \frac{PS} {SQ} = \frac{PT} {TR}$

$\bf\therefore \: ST ∣∣ QR$

$\bf \bf \ \boxed { \bf(If \: a \: line \: divides \: any \: two \: sides \: of \: a \: triangles \: in \: the \: same \: ratio, then \: the \: line \: is \: parallel \: to \: the \: third \: side) }$

$\bf Since \: \: ST ∣∣ QR,$

$\bf\angle \: PST = \angle \: PQR \: \: \: \: (Corresponding angles) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ...(1)$

Also,

$\bf Given \: \angle \: PST = \angle \: PRQ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:...(2)$

From (1) and (2)

$\bf \angle\:PQR = \angle\: PRQ$

(Sides apposite to equal angles are equal)

PR = PQ

Therefore, two sides of △ PQR is equal PQR is an isosceles triangle

Hence proved ✓✓✓