First, we will show that if two circles have the same radius, then they are congruent. To do so, we will start by considering two circles with the same radius, and centers A and B. Let C be a point on the first circle, and D a point on the second circle.
If we can map either ⊙ A or ⊙ B onto the other by using rigid motions, we can conclude that they are congruent. Let's translate ⊙ A to the right so that point A maps onto point B.
Because AC=BD, this translation maps ⊙ A onto ⊙ B. Since translations are rigid motions, we can conclude that ⊙ A≅ ⊙ B.
b In this part, we will prove that if two circles are congruent, then their radii are congruent. Consider the same circles as in Part A.
Given that ⊙ A ≅ ⊙ B, we can conclude that the distance from the center to a point on the circle is the same for both circles. Therefore, since a radius is a segment with endpoints at the center and at any point on a circle, we have that AC ≅ BD.
