Question

Quadrilateral
ABCD is inscribed in circle
O. The
m<B = 4y + 27 and
m<D = y - 12. Determine the
m<D.

Answer 1

Since quadrilateral \(ABCD\) is inscribed in a circle, angles \(B\) and \(D\) are opposite angles. According to the properties of cyclic quadrilaterals, the sum of the measures of opposite angles is \(180^\circ\).

Thus, we have the equation:

\[
m\angle B + m\angle D = 180^\circ
\]

Substituting the given expressions for the angles:

\[
(4y + 27) + (y - 12) = 180
\]

Now, we can combine like terms:

\[
4y + y + 27 - 12 = 180
\]

This simplifies to:

\[
5y + 15 = 180
\]

Next, we can isolate \(y\) by subtracting \(15\) from both sides:

\[
5y = 180 - 15
\]

\[
5y = 165
\]

Now, divide both sides by \(5\):

\[
y = \frac{165}{5} = 33
\]

Now that we have the value of \(y\), we can find \(m\angle D\):

\[
m\angle D = y - 12
\]

Substituting \(y = 33\):

\[
m\angle D = 33 - 12 = 21
\]

Thus, the measure of angle \(D\) is:

\[
\boxed{21^\circ}
\]