Question

In the given figure, $\frac{QR}{QS} = \frac{QT}{PR}$ and $∠1 = ∠2$ . Prove that $△ PQS \sim △ TQR$ .
[Diagram shows triangles PQS and TQR with point S on PR, angle 1 at Q in triangle PQS, angle 2 at Q in triangle TQR.]

Approach

Given $\frac{QR}{QS} = \frac{QT}{PR}$ and $∠1 = ∠2$ . In $△ PQS$ and $△ TQR$ : $∠ PSQ = ∠ TRQ$ (each $9 0^{\circ} - ∠ PQR$ ), $∠ PQS = ∠ TQR$ (given), and the sides around the common angle are proportional. By SAS similarity, $△ PQS \sim △ TQR$ .

Step-by-step working

1
$∠ PSQ = ∠ TRQ$ (each $9 0^{\circ} - ∠ SQR$ )
1 mark
2
$∠ PQS = ∠ TQR$ (given $∠1 = ∠2$ )
0.5 marks
3
$△ PQS \sim △ TQR$ (AA similarity)
0.5 marks

Concepts used

similar trianglessas similarityproportional sides

Approach

Step-by-step working

Concepts used

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