Question:

1. State the given condition – Let \(\triangle ABC\) and \(\triangle DEF\) be two triangles such that\[\angle A = \angle D,\quad \angle B = \angle E,\quad \angle C = \angle F.\]
2. Construct a line parallel to a side – Through point \(D\) draw a line \(\ell\) parallel to side \(BC\) of \(\triangle ABC\). Let \(\ell\) intersect the extension of \(DE\) at \(G\) so that \(DG\) is collinear with \(DE\).
3. Use the parallel line property – Because \(\ell \parallel BC\), the alternate interior angles give\[\angle DGE = \angle B = \angle E,\quad \angle DEG = \angle C = \angle F.\]
4. Identify two triangles with two equal angles – \(\triangle DGE\) and \(\triangle ABC\) have two pairs of equal angles; therefore they are similar (AA similarity). Hence\[\frac{DG}{AB}=\frac{DE}{AC}=\frac{GE}{BC}.\]
5. Relate the sides of the original triangles – Since \(DG\) lies on \(DE\) and \(GE\) lies on \(DF\), the ratios obtained in step 4 reduce to\[\frac{DE}{AB}=\frac{DF}{AC}=\frac{EF}{BC}.\]
6. Conclude the AAA similarity – The three ratios are equal, i.e.,\[\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}.\]Thus the corresponding sides of \(\triangle ABC\) and \(\triangle DEF\) are in the same ratio, proving that the triangles are similar.
7. Result – Hence, if the three corresponding angles of two triangles are equal, the triangles are similar (AAA similarity criterion) and the ratios of their corresponding sides are equal.