If two chords in a circle are congruent, then their central angles are…
Congruent
If two chords in a circle are congruent, then their central angles are also congruent.
This statement is known as the Converse of the Congruent Chords Theorem.
The Congruent Chords Theorem states that if two chords in a circle are congruent, then their corresponding arcs are also congruent. This means that the two chords cut off two arcs of the same length, and these arcs have the same degree measure.
Now, if we draw radii from the center of the circle to the endpoints of each chord, we can see that these radii bisect each chord, dividing them into two segments of equal length. Moreover, each radius also bisects the corresponding arc.
Since the two chords have the same length, and each chord is bisected by a radius, it follows that the two segments of each chord are also congruent. Therefore, the two radii that bisect the chords also have the same length.
Now, if we consider the two triangles formed by drawing radii from the center of the circle to the endpoints of each chord, we can see that they are congruent by the Side-Side-Side (SSS) criterion, since:
– The radii have the same length.
– The chords have the same length (which is half the diameter of the circle).
– The angle between the radii is congruent, since it is equal to half the central angle that subtends the chord.
Therefore, since the two triangles are congruent, it follows that the angles opposite the congruent sides (i.e. the central angles that subtend the chords) are also congruent.
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