Question
Quadrilateral ABCD is inscribed in circle O . The m∠B=4y+27 and m∠D=y−12 . Determine the m∠D .
Answers
GPT-4o mini
To solve for \( m\angle D \) in quadrilateral \( ABCD \) which is inscribed in circle \( O \), we know that the opposite angles of a cyclic quadrilateral sum to \( 180^\circ \). That is:
\[
m\angle B + m\angle D = 180^\circ
\]
Given:
\[
m\angle B = 4y + 27
\]
\[
m\angle D = y - 12
\]
We can set up the equation:
\[
(4y + 27) + (y - 12) = 180
\]
Now, simplify the equation:
\[
4y + 27 + y - 12 = 180
\]
\[
5y + 15 = 180
\]
Next, subtract \( 15 \) from both sides:
\[
5y = 165
\]
Now, divide by \( 5 \):
\[
y = 33
\]
Now that we have found \( y \), we can find \( m\angle D \):
\[
m\angle D = y - 12 = 33 - 12 = 21
\]
Thus, the measure of angle \( D \) is:
\[
\boxed{21}
\]
\[
m\angle B + m\angle D = 180^\circ
\]
Given:
\[
m\angle B = 4y + 27
\]
\[
m\angle D = y - 12
\]
We can set up the equation:
\[
(4y + 27) + (y - 12) = 180
\]
Now, simplify the equation:
\[
4y + 27 + y - 12 = 180
\]
\[
5y + 15 = 180
\]
Next, subtract \( 15 \) from both sides:
\[
5y = 165
\]
Now, divide by \( 5 \):
\[
y = 33
\]
Now that we have found \( y \), we can find \( m\angle D \):
\[
m\angle D = y - 12 = 33 - 12 = 21
\]
Thus, the measure of angle \( D \) is:
\[
\boxed{21}
\]