To find the missing angles in the scenario described, we can use the properties of triangles and the angles in polygons.
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We are given:
- \(m\angle B = 93^\circ\)
- \(m\overparen{BC} = 58^\circ\)
- \(m\overparen{CD} = 106^\circ\)
We need to find \(m\angle A\), \(m\angle C\), and \(m\angle D\).
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Assuming points A, B, C, D form a closed figure (e.g., a polygon), we can analyze their relationships:
- The sum of the angles in any quadrilateral is \(360^\circ\).
- If \( \angle A\), \( \angle B\), \( \angle C\), and \( \angle D\) are the angles of a quadrilateral, then: \[ m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ. \]
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We can solve step by step:
- From our problem, we already have \(m\angle B = 93^\circ\).
- We need to clarify what \(m\angle C\) and \(m\angle D\) are in relation to the segments \(BC\) and \(CD\).
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If \(m\overparen{BC}\) and \(m\overparen{CD}\) indicate the supplementary angles formed at those points (assuming we're forming triangles):
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Let's say \(m\angle C\) is the interior angle facing \(BC\), then it is supplementary to it.
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Therefore, \(m\angle C + m\overparen{BC} = 180^\circ\): \[ m\angle C + 58^\circ = 180^\circ \implies m\angle C = 180^\circ - 58^\circ = 122^\circ. \]
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Similarly, for \(m\angle D\), where it is complementary with \(m\overparen{CD}\): \[ m\angle D + m\overparen{CD} = 180^\circ \implies m\angle D + 106^\circ = 180^\circ \implies m\angle D = 180^\circ - 106^\circ = 74^\circ. \]
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Finally, we can find \(m\angle A\) as follows: \[ m\angle A + 93^\circ + 122^\circ + 74^\circ = 360^\circ. \]
- Adding the known angles: \[ 93^\circ + 122^\circ + 74^\circ = 289^\circ. \]
- So, we solve for \(m\angle A\): \[ m\angle A + 289^\circ = 360^\circ \implies m\angle A = 360^\circ - 289^\circ = 71^\circ. \]
Thus, summarizing the measures:
- \(m\angle A = 71^\circ\)
- \(m\angle C = 122^\circ\)
- \(m\angle D = 74^\circ\)
Final answers: \(m\angle A = \boxed{71^\circ}, m\angle C = \boxed{122^\circ}, m\angle D = \boxed{74^\circ}\).