Question

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)
Responses

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

Answer 1

Since quadrilateral \( AEIU \) is inscribed in circle \( O \), the opposite angles of an inscribed quadrilateral sum up to \( 180^\circ \). We are given the following information:

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

Next, we can find the measures of the angles \( I \) and \( U \) using the property of inscribed quadrilaterals:

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

Substituting \( m \angle A \):

\[
27^\circ + m\angle I = 180^\circ
\]

Solving for \( m \angle I \):

\[
m\angle I = 180^\circ - 27^\circ = 153^\circ
\]

Now, we can find \( m\angle U \) using the angle \( E \):

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

Substituting \( m \angle E \):

\[
54^\circ + m\angle U = 180^\circ
\]

Solving for \( m \angle U \):

\[
m\angle U = 180^\circ - 54^\circ = 126^\circ
\]

Thus, we have:

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

The final answer is:

**m∠I = 153° and m∠U = 126°.**