Question

think about the process if \\( \triangle pqr \\) were similar to \\( \triangle xyz \\), what angle would correspond to \\( \angle p \\)? is \\( \triangle pqr \\) similar to \\( \triangle xyz \\)?

Explanation:

Step1: Calculate the ratios of corresponding sides

For \(\triangle PQR\), assume the lengths of sides. For \(\triangle XYZ\), assume the lengths of sides.
Let's count the units on the coordinate - grid.
Suppose in \(\triangle PQR\), if we assume two sides \(a = 3\) units and \(b= 4\) units. In \(\triangle XYZ\), if two sides \(a'= 1.5\) units and \(b' = 2\) units.
The ratio of the sides is \(\frac{a'}{a}=\frac{1.5}{3}=\frac{1}{2}\) and \(\frac{b'}{b}=\frac{2}{4}=\frac{1}{2}\).
Since the ratios of the corresponding sides are equal.

Step2: Check the angles

Since the triangles are right - angled (by the property of the coordinate - grid, we can assume they are right - angled as they are formed with perpendicular sides on the grid).
For similar triangles, if the ratios of corresponding sides are equal and the included angles (right - angles in this case) are equal.

Answer:

Yes, \(\angle X\)