Proposed Problem
Click the figure below to see the complete problem 382 about Concave Quadrilateral, Angle Bisectors, Angles.
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Complete Geometry Problem 382
Level: High School, SAT Prep
Saturday, October 31, 2009
Problem 382. Concave Quadrilateral, Angle Bisectors, Angles
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Extend CE & CD to meet AB in F & G resply.
ReplyDeleteangle 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
Let alpha=a, beta=b.
ReplyDeleteLet 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))
[For easy typing, I use a for alpha & b for beta]
ReplyDeleteJoin AC
In triangle DAC
b=180-<DAC-<DCA
In triangle EAC
x=180-<DAC-<DCA-<EAD-<ECD
x=b-<EAD-<ECD
<EAD-<ECD=b-x
In triangle BAC
a=180-<DAC-<DCA-<EAD-<ECD-<BAE-<BCE
a=b-(b-x)-(b-x)
a=2x-b
a+b=2x
x=(a+b)/2