Saturday, October 31, 2009

Problem 382. Concave Quadrilateral, Angle Bisectors, Angles

Proposed Problem
Click the figure below to see the complete problem 382 about Concave Quadrilateral, Angle Bisectors, Angles.

 Problem 382. Concave Quadrilateral, Angle Bisectors, Angles.
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Complete Geometry Problem 382
Level: High School, SAT Prep

2 comments:

  1. Extend CE & CD to meet AB in F & G resply.
    angle GFC= α + C/2 and angle x = angle GFC + A/2
    Hence, x = α + C/2 + A/2 or 2(x- α) = A + C
    Likewise, angle AGC = α + C and β = ang. AGC + A or β = α + C + A Or β = α + 2(x- α) = 2x – α or x = (α+ β)/2
    Ajit

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  2. Let alpha=a, beta=b.
    Let measure angle EAD=z and measure angle ECD=w.
    ABCD is a concave quadrilateral,
    hence, b = a + 2z + 2w...(1)
    ABCE is a concave quadrilateral,
    hence, x = z+ w + a...(2)
    (a + b)/2=(a+a+2z+2w)/2...(by (1))
    =2(a+z+w+)/2
    =a+z+w
    =x...(by (2))

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