Question
quadrilateral AEIU is incscribed in circle O. given m angleA = 27 degrees and m angleE = 2 (m angleA), find m angleI and m angleU
Answers
GPT-4o mini
In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the opposite angles are supplementary. This means that the sum of the measures of opposite angles is equal to \(180^\circ\).
Given:
\[
m \angle A = 27^\circ
\]
\[
m \angle E = 2(m \angle A) = 2(27^\circ) = 54^\circ
\]
Now, we can find the measures of angles \(I\) and \(U\):
Since \(m \angle A + m \angle I = 180^\circ\), we have:
\[
m \angle I = 180^\circ - m \angle A = 180^\circ - 27^\circ = 153^\circ
\]
Similarly, since \(m \angle E + m \angle U = 180^\circ\), we have:
\[
m \angle U = 180^\circ - m \angle E = 180^\circ - 54^\circ = 126^\circ
\]
To summarize:
\[
m \angle I = 153^\circ
\]
\[
m \angle U = 126^\circ
\]
Given:
\[
m \angle A = 27^\circ
\]
\[
m \angle E = 2(m \angle A) = 2(27^\circ) = 54^\circ
\]
Now, we can find the measures of angles \(I\) and \(U\):
Since \(m \angle A + m \angle I = 180^\circ\), we have:
\[
m \angle I = 180^\circ - m \angle A = 180^\circ - 27^\circ = 153^\circ
\]
Similarly, since \(m \angle E + m \angle U = 180^\circ\), we have:
\[
m \angle U = 180^\circ - m \angle E = 180^\circ - 54^\circ = 126^\circ
\]
To summarize:
\[
m \angle I = 153^\circ
\]
\[
m \angle U = 126^\circ
\]