4. Quadrilateral ROPQ is inscribed in a circle. Show all work on no credit. Find x and y. What is the measure of each angle of quadrilateral EFGH? What is the measure of arc HF?

Answer:x = 55, y = 60E = 70°, F = 120°, G = 110°, H = 60°arcHF = 140°Step-by-step explanation:a) The sum of opposite angles of an inscribed quadrilateral is 180°. This lets us use angles E and G to solve for x:   (x+15) + (2x) = 180   3x + 15 = 180 . . .simplify   x +5 = 60 . . . . . divide by 3   x = 55 . . . . . . . . subtract 5Similarly, we can use angles F and H to solve for y:   (3y -60) + (y) = 180   4y -60 = 180 . . . . simplify   y -15 = 45 . . . . . . divide by 4   y = 60 . . . . . . . . . add 15___b) Then the measures of the angles are ...   G = 2x = 2·55 = 110   E = 180 -G = 70   H = y = 60   F = 180 -H = 120The angle measures are ...   m∠E = 70°, m∠F = 120°, m∠G = 110°, m∠H = 60°___c) short arc HF is intercepted by inscribed angle E, so the arc will have twice the measure of the angle.   arc HF = 2·m∠E = 140°_____Comment on the problemThroughout, the only relation being used is that the measure of an arc is twice the measure of the inscribed angle intercepting it. For opposite angles of the quadrilateral, the sum of the two intercepted arcs is 360° (the whole circle), so the sum of the two angles is 180°.