Question

quadrilateral ABCD is inscribed in circle O. the m/_B = 4y + 27 and m/_D = y - 1. determine the 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 angles \(B\) and \(D\) is \(180^\circ\).

Given:
\[
m\angle B = 4y + 27
\]
\[
m\angle D = y - 1
\]

Since angles \(B\) and \(D\) are supplementary, we can set up the equation:
\[
m\angle B + m\angle D = 180
\]
Substituting the expressions for \(m\angle B\) and \(m\angle D\):
\[
(4y + 27) + (y - 1) = 180
\]
Combining like terms:
\[
5y + 26 = 180
\]
Next, we solve for \(y\):
\[
5y = 180 - 26
\]
\[
5y = 154
\]
\[
y = \frac{154}{5} = 30.8
\]

Now we can find the measure of angle \(D\):
\[
m\angle D = y - 1 = 30.8 - 1 = 29.8
\]

Therefore, the measure of angle \(D\) is:
\[
\boxed{29.8}
\]