Question

Quadrilateral ABCD is inscribed in circle O . The m∠B=4y+27 and m∠D=y−12 . Determine the m∠D .(1 point) m∠D=

Answer 1

In a cyclic quadrilateral, the opposite angles are supplementary, which means that the sum of the measures of angle B and angle D is 180 degrees. Therefore, we can write the equation:

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

Substituting the expressions for the angles:

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

Now, simplifying the equation:

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

Combining like terms:

\[
5y + 15 = 180
\]

Next, we can isolate \(y\):

\[
5y = 180 - 15
\]
\[
5y = 165
\]
\[
y = \frac{165}{5}
\]
\[
y = 33
\]

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

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

Thus, \(m\angle D\) is \(21\) degrees.

So, the final answer is:

\[
\boxed{21}
\]