Question

In the figure, = QR /QS = QT/PR and ∠1 = ∠ 2 then prove that ∆PQS ~ ∆TQR .

Answer 1

Two Triangles are said to be similar if their i)corresponding angles are equal and ii)corresponding sides are proportional.(the ratio between the lengths of corresponding sides are equal)

•Similarity of triangles should be expressed symbolically using correct correspondence of their vertices.

SOLUTION:

In ΔPQR,
∠1 = ∠2
∠PQR = ∠PRQ         [GIVEN]
∴ PR = PQ ……………..…(1)

[Sides opposite to equal angles of a triangle are also equal]
Given:
QR/QS = QT/PR
QR/QS = QT/PQ

QS/QR = PQ/QT…... …(ii) [Taking reciprocals]

[From eq  (i)]
In ΔPQS and ΔTQR,
QS/QR = PQ/QT             [From eq (ii)]
∠PQS = ∠TQR          [ common]
∴ ΔPQS ~ ΔTQR [By SAS similarity criterion]

Hence, proved

HOPE THIS WILL HELP YOU...

Answer 2

Given : ∠1 = ∠2 and QT/PR = QR/QS.

To prove : ∆PQS ~ ∆TQR

Proof : Since ∠1 = ∠2

→ PR = PQ [Side opp to equal angles are also equal]

Now, QT/PR = QR/QS [Given]

→ QT/PQ = QR/QS

Or

→ PQ/QT = QR/QS [By reciprocal]

→ ∠PQR = ∠TQR = ∠1

Hence by SAS similarity criteria, ∆PQR ~ TQR.

Q.E.D