Theorem
Within a circle or in congruent circles, congruent central angles have congruent chords.
Converse
Within a circle or in congruent circles, congruent chords have congruent central angles.
If
If
You will prove Theorem 12-5 and its converse in Exercises 20 and 36.
Theorem
Within a circle or in congruent circles, congruent chords have congruent arcs.
Converse
Within a circle or in congruent circles, congruent arcs have congruent chords.
If
If
You will prove Theorem 12-6 and its converse in Exercises 21 and 37.
In the diagram,
Why is it important that the circles are congruent?
Two circles may have central angles with congruent chords, but the central angles will not be congruent unless the circles are congruent.
Theorem
Within a circle or in congruent circles, chords equidistant from the center or centers are congruent.
Converse
Within a circle or in congruent circles, congruent chords are equidistant from the center (or centers).
If
If
You will prove the converse of Theorem 12-7 in Exercise 38.