Dilations have some interesting properties that make them unique in geometry. Let's explore these properties:

1. Ratio of Lengths: When a figure is dilated, the ratio of corresponding side lengths remains the same. For example, if a side of the original figure is doubled in length, the corresponding side of the dilated figure will also be doubled.

2. Angle Preservation: Dilations preserve angles. This means that corresponding angles in the original figure and the dilated figure are congruent. For instance, if there is a 90-degree angle in the original figure, the corresponding angle in the dilated figure will also be 90 degrees.

3. Parallel Lines: Dilations retain parallelism. If two lines in the original figure are parallel, their corresponding lines in the dilated figure will also be parallel.

Proof of Properties: These properties can be proved using basic geometric concepts and the definition of dilation. For example, to prove the ratio of lengths property, we can use the fact that dilation stretches or shrinks each dimension equally.

In summary, dilations maintain the ratios of lengths, preserve angles, and preserve parallelism. These properties allow us to easily analyze and work with dilated figures in geometric problems.

Remember, practice makes perfect! Keep exploring different examples of dilations to strengthen your understanding of these key properties.