Question

quadrilateral PQRS is inscribed in circle o. opposite angles p and r measure (x + 16) and (6x - 4), respectively. determine the measure of the smaller angle.
A. 20.
B. 24.
C. 40.
D. 56.

Answer 1

For a quadrilateral inscribed in a circle, opposite angles are supplementary, meaning their measures add up to 180 degrees. In this case, angle \( P \) measures \( (x + 16) \) and angle \( R \) measures \( (6x - 4) \).

We can set up the following equation:

\[
(x + 16) + (6x - 4) = 180
\]

Now, simplify and solve for \( x \):

\[
x + 16 + 6x - 4 = 180
\]
\[
7x + 12 = 180
\]
\[
7x = 180 - 12
\]
\[
7x = 168
\]
\[
x = \frac{168}{7} = 24
\]

Now that we have \( x \), we can find the measures of angles \( P \) and \( R \):

\[
\text{Angle } P = x + 16 = 24 + 16 = 40
\]
\[
\text{Angle } R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140
\]

Since we are looking for the smaller angle, we see that \( 40 \) degrees is the smaller of the two angles.

Thus, the measure of the smaller angle is:

\[
\boxed{40}
\]