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

Answer 1

Since quadrilateral PQRS is inscribed in circle O, the opposite angles are supplementary (add up to 180 degrees).

Therefore, (x + 16) + (6r - 4) = 180.

Combine like terms: x + 6r + 12 = 180.

Now, solve for x: x = 168 - 6r.

Since x is the measure of the smaller angle, we need to find the smallest possible value of x. Since angles in a quadrilateral add up to 360 degrees, the smallest possible value of x will be when the other angles are at their maximum values.

So, if opposite angles P and R are the largest possible values satisfying the conditions, then the smallest angle will be (180 - (x+16)) degrees.

Substituting the expression for x, we get:
180 - (x+16) = 180 - (168 - 6r + 16) = 12 + 6r.

Therefore, the measure of the smaller angle is 12 + 6r degrees.