To analyze the problem involving quadrilateral \(ABCD\) inscribed in circle \(E\), we can apply the properties of cyclic quadrilaterals. In a cyclic quadrilateral, the opposite angles are supplementary; that is, the sum of the measures of opposite angles equals \(180^\circ\).
Given the angle measures:
- \(m\angle A = 54^\circ\)
- \(m\angle B = 115^\circ\)
- \(m\angle C = 86^\circ\)
First, we need to find \(m\angle D\) using the property of opposite angles.
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To find \(m\angle D\): \[ m\angle A + m\angle C = 54^\circ + m\angle D = 180^\circ \] Solving for \(m\angle D\) gives: \[ m\angle D = 180^\circ - 54^\circ = 126^\circ \]
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To find \(m\angle B\): \[ m\angle B + m\angle D = 115^\circ + 126^\circ = 241^\circ \quad (This would imply that angle D should be less than this since this is not possible) \]
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Now we can check the angles:
- The angles opposite to each other sum to \(180^\circ\). Hence, calculating \(D\): \[ 54 + m\angle D = 180 \rightarrow m\angle D = 180 - 54 = 126^\circ \] \[ 115 + m\angle C = 180 \rightarrow m\angle C = 180 - 115 = 65^\circ \]
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Given all remaining checks: \(m\angle C = 65^\circ\)
Now we have to double-check which statement is actually true.
- The measures \(m\angle B\) = \(115\) cannot possibly lead to \(D\) being \(169\), confirming that any of those calculations were based on the angle formulation.
Considering your angular measures' rule for arcs:
- [1]: \(mAD = 115^\circ\) (not possible)
- [2]: \(m\angle 169 = D^\circ\) (not verified)
- [3]: 95.5 = B^\circ (not justified)
- [4]: 84.5 (again not confirmed).
From this deduction and check, it seems as if all statements were flawed, pending a holistic perspective: None stand correct among the detailed evaluations.
Your true statement, based on the correction would be:
- \(mD\) should actually be verified more empirically as (126^\circ*), ruling out all else as misstatements pending.
Double-check each for verification and see where logical consistency is broken.
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