Understanding Similar Triangles

In a right-angled $△ABC$, the length of the hypotenuse $AC=15 \mathrm{cm}$ and $AB=10 \mathrm{cm}$. In a right-angled $△PQR$, the length of the hypotenuse $PR=9 \mathrm{cm}$ and $PQ=6 \mathrm{cm}$. Prove that $△PQR$ and $△ABC$ are similar.

Identify the corresponding vertices if triangles given below are Congruent.

Identify the corresponding vertices if triangles given below are Congruent.

In a right-angled $△PQR$ angled at Q, prove that  $P{Q}^2+Q{R}^2=P{R}^2$

In Fig. $\angle ACB=90° \mathrm{and} CD\perp AB.$

Prove that $\frac{B{C}^2}{A{C}^2}=\frac{{\displaystyle BD}}{AD}$

In the figure, $△PQR$ and $△QRS$ are on the same side of the common base $QR$. Prove that line $SP$ is the perpendicular bisector of $QR$

In the given figure prove that $△\mathrm{GUM}~△\mathrm{BOX}$