Question

In \[{\text{fig}}{\text{.6}}{\text{.36}}\],\[\dfrac{{QR}}{{QS}} = \dfrac{{QT}}{{PR}}\] and \[\angle PQR = \angle PRS\]. Show that \[\Delta PQS \sim \Delta TQR\]
                                                           

Answer 1

Hint: Using the data that corresponding sides of similar polygons or triangle are in proportion, and corresponding angles of similar polygons or triangle have the same measure as both the triangles can be said a similar to each other. So, we just have to verify two triangles in the above diagram whose corresponding sides are in proportion and angle should have the same measures as well.

Complete step by step answer:

From the given diagram we can observe that in \[\Delta PQS\] and \[\Delta TQR\],
As \[\angle PQR = \angle PRS\], so their corresponding sides are also equal as \[PQ = PR\]
And also \[\angle {\text{PQS}} = \angle {\text{TQR}}\], angles are also similar
As per the definition the sides ratio should also be same so,
\[\dfrac{{QR}}{{QS}} = \dfrac{{QT}}{{QP}}\]
These are the ratio of corresponding sides as we know that \[{\text{PQ = PR}}\], and so
\[\dfrac{{QR}}{{QS}} = \dfrac{{QT}}{{PR}}\] which is already mentioned in the question.
Thus, it can be said that the ratio of the sides is also similar and an angle in a triangle is also the same so \[\Delta PQS \sim \Delta TQR\] is shown.

Note: If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion.