Question

1. what is our definition of similarity?

a. if 2 figures have the same angles, then they are similar.
b. if 2 figures have proportional side lengths, then they are similar.
c. if there is a sequence of rigid transformations taking one figure to another, then they are similar.
d. if there is a sequence of rigid transformations and dilations that take one figure to the other, then they are similar.

Explanation:

Step1: Analyze Option A

Just having the same angles is not sufficient. For example, a square and a rectangle can have all angles equal (90 degrees) but not be similar as their side - length ratios can be different.

Step2: Analyze Option B

Proportional side lengths alone are not enough. If the angles are not equal, the figures are not similar. For instance, a rhombus (all sides equal - proportional in a sense) and a square (all sides equal) are not similar if the rhombus has non - right angles.

Step3: Analyze Option C

Rigid transformations (translations, rotations, reflections) preserve the size and shape of a figure. So, if there is only a sequence of rigid transformations taking one figure to another, the figures are congruent (a special case of similarity with a scale factor of 1), but the general definition of similarity includes non - congruent cases.

Step4: Analyze Option D

By the definition of similarity in geometry, two figures are similar if one can be obtained from the other by a sequence of rigid transformations (to re - orient and re - position) and a dilation (to change the size while keeping the shape, i.e., angle measures the same and side - length ratios proportional).

Answer:

D. If there is a sequence of rigid transformations and dilations that take one figure to the other, then they are similar.