Question

given: ⊙o with central angles ∠aoc ≅ ∠bod
prove: \\(\overline{ac} \cong \overline{bd}\\)

complete the missing parts of the paragraph proof.
proof:
we know that central angles aoc and bod are congruent, because it is given. we can say that segments ao, co, bo, and do are congruent because dropdown menu. then by the reason we know that triangle triangle finally, we can conclusion to chord bd dropdown menu.
dropdown options for the first blank: all radii of a circle are congruent, of the definition of central angles, of the isosceles triangle theorem
image of a circle with center o, points a, b, c, d on the circumference, with chords ac, bd, and radii ao, bo, co, do drawn

Explanation:

To determine why segments \( AO \), \( CO \), \( BO \), and \( DO \) are congruent, we recall a fundamental property of circles: all radii of a circle are congruent. In circle \( \odot O \), \( AO \), \( CO \), \( BO \), and \( DO \) are all radii (since \( A \), \( B \), \( C \), \( D \) lie on the circle and \( O \) is the center). So the reason for their congruence is that all radii of a circle are congruent.

For the next part (though the full context of the "then by the..." and "Finally..." is a bit truncated, the key here is the first missing reason). The other options: "of the definition of central angles" doesn't explain radius congruence, and "of the isosceles triangle theorem" is about triangle sides/angles, not radii. So the correct choice for the first dropdown (reason for \( AO \cong CO \cong BO \cong DO \)) is "all radii of a circle are congruent".

Answer:

all radii of a circle are congruent