Question

which statement is true?
△pqr is similar to △rst;
△pqr is not similar to △rst;
there is not enough information to determine whether the triangles are similar.

Explanation:

Step1: Check the right angles

Both \(\triangle AQR\) and \(\triangle RST\) have a right - angle (\(\angle AQR = \angle RST=90^{\circ}\))

Step2: Check the ratio of sides

Let's assume the ratio of the corresponding sides. If we consider the sides adjacent to the right - angles. Let's say in \(\triangle AQR\) and \(\triangle RST\), if we assume the ratio of the sides \( \frac{AQ}{RS}\) and \( \frac{AR}{RT}\). But we know that for two right - angled triangles \(\triangle AQR\) and \(\triangle RST\), if \(\angle AQR=\angle RST = 90^{\circ}\) and \(\angle QAR+\angle AQR+\angle A RQ=180^{\circ}\), \(\angle SRT+\angle RST+\angle STR = 180^{\circ}\). Also, \(\angle A RQ+\angle SRT = 90^{\circ}\) (since they form a linear pair at point \(R\)). But we can use the AA (Angle - Angle) similarity criterion. Since both triangles are right - angled (\(\angle AQR=\angle RST = 90^{\circ}\)) and \(\angle QAR\) and \(\angle SRT\) are complementary to the same angle (the non - right angle at \(R\) for the two triangles). So, \(\angle QAR=\angle STR\) (by the property of complementary angles).

Answer:

\(\triangle AQR\) is similar to \(\triangle RST\)