Quadrilateral AEIU is inscribed in circle O . Given m∠A=27° and m∠E=2(m∠A) , find m∠I and m∠U .(1 point)

Question

Responses

m∠I=126° and m∠U=126°
m angle upper I equals 126 degrees and m angle upper U equals 126 degrees

m∠I=153° and m∠U=153°
m angle upper I equals 153 degrees and m angle upper U equals 153 degrees

m∠I=153° and m∠U=126°
m angle upper I equals 153 degrees and m angle upper U equals 126 degrees

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 start with the given information:

1. \( m\angle A = 27^\circ \)
2. \( m\angle E = 2(m\angle A) = 2(27^\circ) = 54^\circ \)

In any cyclic quadrilateral (one inscribed in a circle), the opposite angles sum to \( 180^\circ \). This gives us the following relationships:

\[ m\angle A + m\angle I = 180^\circ \] \[ m\angle E + m\angle U = 180^\circ \]

Now substituting the known angles into these equations:

1. From \( m\angle A + m\angle I = 180^\circ \): \[ 27^\circ + m\angle I = 180^\circ \] \[ m\angle I = 180^\circ - 27^\circ = 153^\circ \]

2. From \( m\angle E + m\angle U = 180^\circ \): \[ 54^\circ + m\angle U = 180^\circ \] \[ m\angle U = 180^\circ - 54^\circ = 126^\circ \]

Thus, we have found the measures of the angles:

• \( m\angle I = 153^\circ \)
• \( m\angle U = 126^\circ \)

Therefore, the correct answer is:

\( m\angle I = 153^\circ \) and \( m\angle U = 126^\circ \).