Introduction to Congruence and Similarity

In geometry, transformations help us compare the shapes and sizes of different figures. By sliding, turning, flipping, or resizing shapes, we can determine if they are congruent or similar.

What is Congruence?

Two figures are congruent if they have the exact same shape and the exact same size.

You can think of congruent figures as perfect copies of one another. You can map one figure exactly onto the other using only rigid transformations. These include:

• Translations: Sliding the figure.
• Reflections: Flipping the figure over a line.
• Rotations: Turning the figure around a point.

Because these transformations do not change the size of the figure, all corresponding angles and corresponding side lengths remain completely equal. The mathematical symbol for congruence is $\cong$.

What is Similarity?

Two figures are similar if they have the same shape, but not necessarily the same size.

Similar figures are created when you use a dilation (resizing by shrinking or enlarging), often combined with translations, reflections, or rotations. For any two similar figures, two rules always apply:

1. Corresponding angles are perfectly equal.
2. Corresponding side lengths are proportional (they have the same ratio).

The mathematical symbol for similarity is $\sim$.

Example Problems

Example 1: Identifying Congruence

Question: Triangle A has side lengths of $5$, $7$, and $9$. Triangle B has side lengths of $5$, $9$, and $7$. Are they congruent?

Answer: Yes! Even though the order of the sides is listed differently, both triangles have the exact same three side lengths. This means Triangle A can be rotated or reflected to perfectly match Triangle B. Because no resizing is needed, the triangles are congruent.

Example 2: Using Similarity Ratios

Question: Two similar triangles have corresponding sides in a ratio of $2:3$. If a side on the smaller triangle has a length of $6$, what is the length of the corresponding side on the larger triangle?

Answer: Since the triangles are similar, their sides are proportional. We can set up a proportion using the given ratio:

$\frac{2}{3} = \frac{6}{x}$

To solve for $x$ (the missing side), we can cross-multiply:

$2x = 3 \times 6$

$2x = 18$

$x = 9$

The corresponding side on the larger triangle is $9$.