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 similarity

The corresponding angles must be equal. That is, $\angle A=\angle D$, $\angle B = \angle E$, and $\angle C=\angle F$. This is known as the AA (angle - angle) similarity criterion.

Step2: Side - side - side similarity

The ratios of the corresponding sides must be equal. So, $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}=k$, where $k$ is the scale factor. This is the SSS (side - side - side) similarity criterion.

Step3: Side - angle - side similarity

If two pairs of corresponding sides are in proportion and the included angles are equal. For example, $\frac{AB}{DE}=\frac{AC}{DF}$ and $\angle A=\angle D$. This is the SAS (side - angle - side) similarity criterion.

Answer:

The corresponding angles must be equal ($\angle A=\angle D$, $\angle B=\angle E$, $\angle C=\angle F$) and the ratios of the corresponding sides must be equal ($\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$), or two pairs of corresponding sides are in proportion and the included angles are equal.