Question

in the given figure if angle tqr+angle stq=180 , ps=2cm, sr=3cm and qr=10 cm , then find the length of ts. ​

Answer 1

The length of TS is 4 cm

Step-by-step explanation:

Given

$\angle TQR+\angle STQ=180^\circ$

$PS=2$ cm

$SR=3$ cm

$QR=10$ cm

Since $\angle TQR$ and $\angle STQ$ are on the same side of lines TS and QR cut by a line PQ

Therefore, $TS\parallel QR$

$\therefore, \angle PTS=\angle TQR$    (corresponding angles)

We can write $\angle TQR$ as $\angle PQR$

Thus,

$\angle PTS=\angle PQR$

Similarly,

$\angle PST=\angle PRQ$        (corresponding angles)

In $\triangle PTS \text{ and } \triangle PQR$

$\angle P=\angle P$     (common to both the triangles)

$\angle PTS=\angle PQR$

$\angle PST=\angle PRQ$  

$\therefore \triangle PTS\sim \triangle PQR$

And the sides of similar triangles are proportional

Therefore,

$\frac{PS}{PR}=\frac{TS}{QR}=\frac{PT}{PQ}$

$\implies \frac{PS}{PR}=\frac{TS}{QR}$

$\implies \frac{2}{2+3}=\frac{TS}{10}$

$\implies TS=\frac{2}{5}\times 10$

$\implies TS=4$ cm

Hope this answer is helpful.

Know More:

Q: In the given figure, angle OEB =75°, angle OBE =55°, and angle OCD =100°. Find angle ODC (with proper steps).​

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