Question

Quadrilateral PQRS is inscribed inside circle O. The opposite angles are P and R. Angle P measures (x+10) degrees and angle R measures (5x-28) degrees respectively. Determine the measure of angle R. Show all work in order to receive full credit. You may upload a picture of your work using the picture button or type your answer in. (3 points)

Answer 1

To solve for the measure of angle R in the inscribed quadrilateral PQRS, let's use the property that in any cyclic quadrilateral (a quadrilateral inscribed in a circle), the sum of opposite angles is equal to 180 degrees.

Given that:
- Angle P = \( (x + 10) \) degrees
- Angle R = \( (5x - 28) \) degrees

According to the property mentioned above, we can set up the equation:

\[
\text{Angle P} + \text{Angle R} = 180
\]
Substituting the expressions for angle P and angle R:

\[
(x + 10) + (5x - 28) = 180
\]

Now, we can simplify the equation:

\[
x + 10 + 5x - 28 = 180
\]

Combining like terms, we get:

\[
6x - 18 = 180
\]

Next, we add 18 to both sides to isolate the term with x:

\[
6x = 198
\]

Now, we divide by 6 to solve for x:

\[
x = 33
\]

Now that we have the value of x, we can substitute it back into the expression for angle R:

\[
\text{Angle R} = 5x - 28
\]

Substituting \( x = 33 \):

\[
\text{Angle R} = 5(33) - 28
\]
\[
\text{Angle R} = 165 - 28
\]
\[
\text{Angle R} = 137
\]

Thus, the measure of angle R is:

\[
\boxed{137 \text{ degrees}}
\]