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Rule

Congruent Circles Theorem

Two circles are congruent circles if and only if they have the same radius.
Two circles with a radius drawn on each

Based on the above diagram, the theorem can be written as follows.

Proof

The biconditional statement will be proved separately.

If Two Circles Are Congruent, Then They Have the Same Radius

Consider two congruent circles and and a point on each one.

Two circles with a point drawn on each one

Because the distance from the center to a point on the circle is the same for both circles. Therefore, which implies that both circles have the same radius. That is,

Two circles and a radius drawn on each one. The two radii are congruent.

It has been proved that if two circles are congruent, then they have the same radius.

If Two Circles Have the Same Radius, Then They Are Congruent

Consider now two circles with the same radius.

Two circles and a radius drawn on each one. The two radii are congruent.
Next, can be translated so that point is mapped onto point The image of is which is a circle centered at Since the circles have the same radius, this translation maps onto
Two circles with the same radius. One of them is translated onto the other.
Because a rigid motion maps one circle onto the other, it is concluded that both circles are congruent.