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

Since quadrilateral ABCD is inscribed in circle O, it is a cyclic quadrilateral. For cyclic quadrilaterals, the opposite angles are supplementary. This means that:

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

Given the measures of the angles:

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

We can set up the equation:

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

Now, simplify the left side:

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

Next, we can solve for \(y\):

\[
5y = 180 - 15
\]
\[
5y = 165
\]
\[
y = 33
\]

Now, substitute \(y\) back into the expression for \(m\angle D\):

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

Therefore, the measure of angle D is:

\[
\boxed{21}
\]