1. what must be true in order for triangles def and abc to be similar? be very specific about the properties…

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.