quadrilateral PQRS is inscribed in circle O. Opposite angles P and R measure (x+16) degrees and (6r-4) degrees, respectively. determine the measure of the smaller angle

Question

the answer is either 40 degrees,24 degrees,20 degrees or 56 degrees

Answer 1

Opposite angles in an inscribed quadrilateral are supplementary, meaning they add up to 180 degrees.

Thus, we have:
(x + 16) + (6r - 4) = 180

Simplifying the equation, we get:
x + 6r + 12 = 180

We also know that the sum of all angles in a quadrilateral is 360 degrees, so we have:
(x + 16) + (6r - 4) + 90 + 90 = 360

Simplifying the equation, we get:
x + 6r + 192 = 360

From the two equations we get:
x + 6r + 192 = 360
x + 6r + 12 = 180

Subtracting the second equation from the first equation, we get:
180 - 12 = 360 - 192
168 = 168

This means that the equations are consistent and all solutions will be valid.

Now, to find the smaller angle, we can substitute back into the first equation:
(x + 16) = 180 - (6r - 4)
(x + 16) = 184 - 6r

Plugging in different values of r, we find that the smaller angle can be:
56 degrees (r = 4) when x = 40
40 degrees (r = 3) when x = 24
24 degrees (r = 2) when x = 22
20 degrees (r = 1) when x = 18

Therefore, the answer could be 56 degrees, 40 degrees, 24 degrees, or 20 degrees, depending on the values of x and r.