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 (a quadrilateral inscribed in a circle), the opposite angles are supplementary. This means that the sum of the measures of angle \( B \) and angle \( D \) is \( 180^\circ \).

Given:
- \( m\angle B = 4y + 27 \)
- \( m\angle D = y - 12 \)

According to the properties of cyclic quadrilaterals:

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

Substituting the expressions for \( m\angle B \) and \( m\angle D \):

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

Now, combine like terms:

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

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

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

Now, divide by 5:

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

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

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

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

\[
\boxed{21}
\]