Question

quadrilateral ABCD is inscribed in circle O. the m angleB = 4y + 27 and m angleD = y - 12. determine the m angleD

Answer 1

Since quadrilateral ABCD is inscribed in circle O, we know that the opposite angles of an inscribed quadrilateral are supplementary. That means:

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

Given:

\[
m \angle B = 4y + 27
\]

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

We substitute the expressions for \(m \angle B\) and \(m \angle D\) into the equation:

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

Now, combine like terms:

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

This simplifies to:

\[
5y + 15 = 180
\]

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

\[
5y = 180 - 15
\]

\[
5y = 165
\]

Now, divide by 5:

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

Now that we have \(y\), we can determine \(m \angle D\):

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

Therefore, the measure of angle D is:

\[
\boxed{21}
\]