Given that quadrilateral \( AEIU \) is cyclic (inscribed in circle \( O \)), we can use the property that opposite angles in a cyclic quadrilateral sum to \( 180^\circ \).
From the problem, we know:
- \( m\angle A = 27^\circ \)
- \( m\angle E = 2(m\angle A) = 2 \times 27^\circ = 54^\circ \)
Let:
- \( m\angle I = x \)
- \( m\angle U = y \)
According to the properties of cyclic quadrilaterals: \[ m\angle A + m\angle I = 180^\circ \] \[ m\angle E + m\angle U = 180^\circ \]
Substituting the known values into these equations:
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From \( m\angle A + m\angle I = 180^\circ \): \[ 27^\circ + x = 180^\circ \implies x = 180^\circ - 27^\circ = 153^\circ \]
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From \( m\angle E + m\angle U = 180^\circ \): \[ 54^\circ + y = 180^\circ \implies y = 180^\circ - 54^\circ = 126^\circ \]
Thus the measures of the angles are: \[ m\angle I = 153^\circ \quad \text{and} \quad m\angle U = 126^\circ \]
In summary: \[ m\angle I = 153^\circ \quad \text{and} \quad m\angle U = 126^\circ \]
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