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# Dear smart,

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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

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

HandLoin  Nov 13, 2017
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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

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