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Quadrilateral PQRS is circumscribed about a circle. Line PQ is tangent to the circle at A, line QR at B, line RS at C, and line PS at D. If measure of angle P = 68, measure of angle Q = 74, and measure of angle R = 114, find the measures of the angles of quadrilateral ABCD.

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The sum of the angles of a quadrilateral is 360 degrees.  So P+Q+R is 206, 360 - 206 = 154 degrees, the measure of angle S.

The four triangles, AQB, BRC, CSD, and DPA are all isosceles.  So angle QBA = angle BAQ, etc.  We find QBA = (180-24)/2 or 78 degrees.

RBC = (180-114)/2 = 33 degrees.

180 - (78 + 33) is the measure of angle B:  69 degrees.

The student should be able to see how to calculate the missing information from this.

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