Which Circles Are Congruent?

In the plane, all circles of the same radius are congruent to one another. In this case, radius AB is of the same length as the radius DC.

Here is a great video on congruent circles and their properties!

Geometrically, congruence is a property of at least two shapes relative to each other. If two shapes are congruent, then they share the exact same properties. Furthermore, congruence is not related to where the shapes are, how they are rotated, or how they are reflected, since none of these changes affect the size of the shape.

We've seen before that a circle is defined by two properties: its center and radius. Since congruence is independent of position (and circles remain the same regardless of rotation or reflection), the congruence of circles depends only on the circle radii.

But wait! Since a radius is a constant (an unchanging number), and any constant is only equal to another constant when they are equal, then all circles with the same radii (or values dependent on radii) must be congruent.

A related geometric property is similarity. Similar shapes are proportional in every possible way. Like congruence, similarity is unaffected by position, rotation, or reflection.

All congruent shapes are similar by definition; however, not all similar shapes are congruent.

We said earlier that congruent circles are exactly identical. This property has some interesting consequences.

Since congruent circles can be defined by their congruent radii, properties of circles that are determined by a radius must also be congruent.

Thus, the circumferences, diameters, and arclengths (with the same angles) are also congruent in congruent circles!