Question

In Figure, $$\triangle PQR$$ is right angled at $$P$$, $$U$$ and $$T$$ are the points on line $$QR$$, If $$QP \parallel ST$$ and $$US \parallel RP$$, find $$\angle S$$.

Answer 1

$$QP$$ parallel to $$ST$$ and $$QT$$ is a transversal.

$$\therefore$$ $$\angle PQR = \angle STU$$ .... Alternate interior angles

$$US$$ parallel to $$PR$$ and $$UR$$ is a transversal.

$$\therefore$$ $$\angle PRQ = \angle SUT$$ ....Alternate interior angles

Here, two angles are equal to the two angles of another triangle. Hence, the third one will also equal.

Also, by $$AA$$ test, $$\Delta PQR\sim \Delta STU$$

$$\therefore$$ $$\angle QPR = \angle TSU$$

Now, given that $$\angle P = 90^{o}$$

$$\therefore$$ $$\angle S = 90^{o}$$