Question

Quadrilateral AEIU is inscribed in circle O. Given m∠A = 27° and m∠E = 2 (m∠A), find m∠I and m∠U.


a. m∠I = 153° and m∠U = 126°
b. m∠I = 126° and m∠U = 126°
c. m∠I = 153° and m∠U = 153°
d. m∠I = 126° and m∠U = 153°

Answer 1

To solve for \( m\angle I \) and \( m\angle U \) in the inscribed quadrilateral \( AEIU \), we will use the property of cyclic quadrilaterals that states opposite angles are supplementary.

1. We are given:
- \( m\angle A = 27^\circ \)
- \( m\angle E = 2 \times m\angle A = 2 \times 27^\circ = 54^\circ \)

2. Since \( AEIU \) is a cyclic quadrilateral, we know:
\[
m\angle A + m\angle I = 180^\circ
\]
\[
m\angle E + m\angle U = 180^\circ
\]

3. To find \( m\angle I \):
\[
m\angle I = 180^\circ - m\angle A = 180^\circ - 27^\circ = 153^\circ
\]

4. To find \( m\angle U \):
\[
m\angle U = 180^\circ - m\angle E = 180^\circ - 54^\circ = 126^\circ
\]

Thus, we have:
- \( m\angle I = 153^\circ \)
- \( m\angle U = 126^\circ \)

The answer is: **a. \( m\angle I = 153^\circ \) and \( m\angle U = 126^\circ \)**.