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