Question

In circle q, ∠rqs ≅ ∠sqt. Circle q is shown. Line segments qr, qs, qt, qu, and qu are radii. Lines are drawn to connect the point on the circle to create secants uv, vr, rs, st, and tu. Which statement must be true?

1) Arc rt ≅ Arc ut
2) ∠rqt ≅ ∠rst
3) rq ⊥ qt
4) rs ≅ st

Answer 1

Based on the properties of circle q with congruent angles and secants, the correct statements are that Arc rt ≅ Arc ut (option 1) and rs ≅ st (option 4), because congruent angles intercept congruent arcs and by definition, all radii in the same circle are equal.

The correct answer is 1).

Step-by-step explanation:

The question involves the properties of circle q, where it is given that ∠rqs ≅ ∠sqt. With qr, qs, qt, and qu being radii of the circle and secants uv, vr, rs, st, and tu connecting points on the circle. To find which statement must be true based on these conditions, we examine each option:

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• 1) Arc rt ≅ Arc ut: Given that ∠rqs ≅ ∠sqt, the intercepted arcs rt and ut would also have to be congruent.
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• 2) ∠rqt ≅ ∠rst: This cannot be concluded without additional information on the angles or other aspects of the shape.
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• 3) rq ∥ qt: This can only be concluded if we know that the shape is a right angle, which is not provided in the statement.
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• 4) rs ≅ st: This is true by definition, as they are both radii of the same circle.

The correct statements based on the information provided are that Arc rt ≅ Arc ut and rs ≅ st because they are congruent arcs intercepted by congruent angles and congruent radii, respectively. Option 1 and option 4 must be true.