Question

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

m∠D = ___°

Answer 1

Since quadrilateral ABCD is inscribed in a circle (cyclic quadrilateral), the opposite angles of a cyclic quadrilateral sum up to 180 degrees. Therefore, we can write the equation for angles B and D as follows:

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

We are given:

\[
\text{m}\angle B = 4y + 27
\]
\[
\text{m}\angle D = y - 12
\]

Substituting these expressions into the equation:

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

Combining like terms:

\[
4y + y + 27 - 12 = 180
\]
\[
5y + 15 = 180
\]

Now, solve for \(y\):

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

Now that we have \(y\), we can find m∠D:

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

Thus, the measure of angle D is:

\[
\text{m}\angle D = 21^\circ
\]

So, the final answer is:

\[
\boxed{21}
\]