Question

Quadrilateral PQRS is inscribed in circle 0. Opposite angles P and R measure (x + 16) • and (6x - 4) °, respectively. Determine the measure of the smaller angle.
(1 point)
020°
0 40°
0 24°
0 56°

Answer 1

In a cyclic quadrilateral (one that is inscribed in a circle), the opposite angles are supplementary. This means that the sum of the measures of angles P and R is equal to 180°.

Given:
- Angle P = \(x + 16\)
- Angle R = \(6x - 4\)

We can set up the equation based on the property of opposite angles in a cyclic quadrilateral:

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

Now, let's simplify this equation:

\[
x + 16 + 6x - 4 = 180
\]
\[
7x + 12 = 180
\]

Next, subtract 12 from both sides:

\[
7x = 180 - 12
\]
\[
7x = 168
\]

Now, divide both sides by 7 to solve for \(x\):

\[
x = \frac{168}{7} = 24
\]

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

For angle P:
\[
P = x + 16 = 24 + 16 = 40^\circ
\]

For angle R:
\[
R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140^\circ
\]

Therefore, the measures of angles P and R are:
- Angle P = 40°
- Angle R = 140°

To find the smaller angle, we look at the measures of the angles:
- The smaller angle of the two is \(40^\circ\).

Thus, the answer is \( \boxed{40^\circ} \).