Question

1. what must be true in order for triangles def and abc to be similar? be very specific about the properties of the angles and sides of these two triangles.

Explanation:

Step1: Angle - Angle (AA) Similarity

For two triangles to be similar, the corresponding angles must be equal. So, \(\angle A=\angle D\), \(\angle B = \angle E\), and \(\angle C=\angle F\).

Step2: Side - Angle - Side (SAS) Similarity

If the ratios of the lengths of two pairs of corresponding sides are equal and the included angles are equal. Let \(\frac{AB}{DE}=\frac{BC}{EF}\) and \(\angle B=\angle E\) (or \(\frac{AB}{DE}=\frac{AC}{DF}\) and \(\angle A=\angle D\) or \(\frac{BC}{EF}=\frac{AC}{DF}\) and \(\angle C=\angle F\)).

Step3: Side - Side - Side (SSS) Similarity

The ratios of the lengths of all three pairs of corresponding sides are equal. That is \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\).

Answer:

Triangles \(DEF\) and \(ABC\) are similar if either:

• All corresponding angles are equal (\(\angle A=\angle D\), \(\angle B=\angle E\), \(\angle C = \angle F\)) (by AA similarity).
• The ratios of two pairs of corresponding sides are equal and the included angles are equal (e.g., \(\frac{AB}{DE}=\frac{BC}{EF}\) and \(\angle B=\angle E\)) (by SAS similarity).
• The ratios of all three pairs of corresponding sides are equal (\(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\)) (by SSS similarity).