To solve this problem, we'll use the properties of a cyclic quadrilateral and the measures of the arcs that subtend the angles.
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Understanding Arc Measures:
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In a cyclic quadrilateral, the measure of an angle is half the measure of the intercepted arc.
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The angle \( \angle A \) is subtended by arcs \( BC \) and \( CD \), so: \[ m\angle A = \frac{1}{2}(m \text{ arc } BC + m \text{ arc } CD) = \frac{1}{2}(115 + 86) = \frac{1}{2}(201) = 100.5^\circ \]
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The angle \( \angle B \) is subtended by arcs \( CD \) and \( DA \), so we need to find \( m \text{ arc } DA \).
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The total measure of arcs in the circle is 360 degrees, so: \[ m \text{ arc } AD = 360 - (m \text{ arc } AB + m \text{ arc } BC + m \text{ arc } CD) \] Calculating \( m\text{ arc } AB \): \[ m\text{ arc } AD = 360 - (54 + 115 + 86) = 360 - 255 = 105 \]
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Then we can find angle \( B \): \[ m\angle B = \frac{1}{2}(m \text{ arc } CD + m \text{ arc } AD) = \frac{1}{2}(86 + 105) = \frac{1}{2}(191) = 95.5^\circ \]
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The angle \( \angle C \): \( m\angle C \) is subtended by arcs \( DA \) and \( AB \): \[ m\angle C = \frac{1}{2}(m \text{ arc } DA + m \text{ arc } AB) = \frac{1}{2}(105 + 54) = \frac{1}{2}(159) = 79.5^\circ \]
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The angle \( D \): \( m\angle D \) is subtended by arcs \( AB \) and \( BC \): \[ m\angle D = \frac{1}{2}(m \text{ arc } AB + m \text{ arc } BC) = \frac{1}{2}(54 + 115) = \frac{1}{2}(169) = 84.5^\circ \]
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Evaluating the Statements Given:
- A. \( m<B = 95.5 \) — True
- B. \( m<D = 169 \) — False (It is not 169, it's 84.5)
- C. \( m \text{ arc } AD = 115 \) — False (It was calculated as 105)
- D. \( m \text{ arc } ADC = 84.5 \) — False (This is not the correct arc measure)
Overall, the true statement is:
- A. \( m<B = 95.5 \).
1 answer