To explain why the word congruent is essential for both theorems, we will begin by investigating the theorems.
Theorem 12-4: Within a circle or in congruent circles, congruent central angles have congruent arcs.
Theorem 12-5: Within a circle or in congruent circles, congruent central angles have congruent chords.
Let's visualize the theorems for the congruent circles. Recall that the corresponding parts of congruent circles are congruent. Therefore, we can conclude that the radii of congruent circles are congruent.
By these theorems, if ∠ AO_1B ≅ ∠ CO_2D, then AB ≅ CD and AB ≅ CD. Now, let's consider the length of an arc.
arc length=2π r * α/360^(∘)
As we can see, the arc length is determined not only by the central angle but also by the radius of the circle. Next, we will consider the length of a chord. The length of a chord can be found by applying the Law of Cosines.
chord length=sqrt(2 r^2-2 r^2cos α)
The length of a chord is also determined by both the central angle and the radius of a circle. Therefore, for the corresponding arcs and chords to be congruent, the circles must be congruent.
⊙ O_1 ≅ ⊙ O_2
Otherwise, even if the central angles are congruent, the corresponding arcs and chords wouldn't be congruent. This is why these theorems must specify that the circles are congruent.
