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Since the definition of diameter tells us that a diameter is a special kind of chord, we know that any diameter is also a chord. So the statement is true. The converse, on the other hand, is not true, since not all chords are diameters.
Any radius is also a chord. True or false?
A radius is by definition a line segment that has only one of its endpoints on the circle. Therefore, no radius of a given circle can be a chord of the same circle. The statement is false.
In the figure below, points A, B, C, and D are on ⊙O and AB ≅ CD. Do we know for sure whether arc AB is congruent to arc CD?
AB and CD are chords of ⊙O. Since any circle is congruent to itself, the Arc-Chord Theorem tells us that if two chords are congruent, then their associated arcs are congruent. Therefore we know for sure that arc AB is congruent to arc CD.
In the figure below, points A, B, C, and D are on ⊙O, m∠AOB = 37°, and m∠DOC = 40°. Do we know for sure whether AB ≅ CD?
They're not drawn in the picture, but we know AB and CD are chords of ⊙O. Since any circle is congruent to itself, the Angle-Chord Theorem totally applies. It tells us that two chords are congruent if and only if their associated central angles are congruent. But the problem tells us that ∠AOB is not congruent to ∠COD, because they have different measures. Therefore, we know for sure that AB is not congruent to CD. Bummer.