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In the diagram, quadrilateral ABCD is inscribed in the circle, arc ADB is a minor arc, and segment AB is parallel to segment DC. Given that arc DC is 30 degrees, arc AD is x^2 + 7x degrees, and arc BC is 60 - 4x degrees, find the measure of arc AEB.

https://www.mediafire.com/file/sovz5e8xlsisje1/Screen%20Shot%202017-11-12%20at%208.33.03%20PM.png

(download) (no virus i swear the url says png!!)

\([asy] unitsize(2 cm); pair A, B, C, D, E; A = dir(170); B = dir(30); D = dir(130); C = dir(70); E = dir(280); draw(Circle((0,0),1)); draw(A--B--C--D--cycle); label("$A$", A, W); label("$B$", B, dir(0)); label("$C$", C, NE); label("$D$", D, NW); dot("$E$", E, S); [/asy]\)

HandLoin Nov 13, 2017

#1**+1 **

Since AB is parallel to BC, then chords AD and BC are equal....thus

x^2 + 7x = 60 - 4x rearrange as

x^2 + 11x - 60 = 0 factor

(x - 4) ( x + 15) = 0 setting each factor to 0 and solving for x we have that

x = 4 or x = -15

Taking the positive solution, then BC = AB = 60 - 4(4) = 60 - 16 = 44

Thus arcs AD + DC + CB = [ 44 + 30 + 44 ] = 118°

Thus arc AEB = 360 - 118 = 242°

CPhill Nov 13, 2017