Arcs and Chords
Speaking of friends, let's return (notice we refrained from saying "circle back") to our discussion of friendships. We know that arcs are great friends with central angles. We just learned that chords are also great friends with central angles. It's not too far-fetched to imagine a central angle throwing a party and inviting both its associated chord and its associated arc. Who's bringing the hummus?
Indeed, since each central angle "knows" an arc and a chord, it can form the bridge of friendship between those two—introducing them to each other, making sure they have plenty to talk about, overall making sure they get to know each other and are having a good time at the party.
An arc and a chord that share a central angle ought to get along just fine. After all, they have two points in common.
In fancy talk, two chords are congruent if and only if their associated arcs are congruent. We've got another biconditional here, and you know what that means: we have to prove both directions of the statement.
In one way, if two chords are congruent, then their associated arcs are congruent. In the other, if two arcs are congruent, then their associated chords are congruent.
Given that ⊙O is congruent to ⊙O' with chords AB and CD, we can start by drawing in some extra line segments: OA, OB, O'C, and O'D.
These segments form the central angles of chords AB and CD: ∠AOB and ∠CO'D, respectively. We are given that AB is congruent to CD. Since the chords are congruent, we know that ∠AOB and ∠CO'D are congruent as well. Since arcs AB and CD have the same measure and are segments of congruent circles, we can say that arcs AB and CD are congruent by definition of congruent arcs.
To prove that AB is congruent to CD given that arc AB is congruent to arc CD, we can use pretty much the same logic we used, but in reverse. Here, we're given that arcs AB and CD are congruent. Then by definition of congruent arcs, we know that their associated central angles, ∠AOB and ∠CO'D, are also congruent. That means chords AB and CD are congruent.
We'll call this relationship the Arc-Chord Theorem. And so begins a glorious friendship between the arc and the chord.