Geometry : 116

22. Isosceles angle

Let ABC be an isosceles triangle (AB = AC) with angle BAC = 20°.  Point D is on side AC such that DBC = 60°.  Point E is on side AB such that ECB = 50°.  Find, with proof, the measure of EDB.

 

 

Solution to puzzle 22: Isosceles angle

Let ABC be an isosceles triangle (AB = AC) with angle BAC = 20°.  Point D is on side AC such that DBC = 60°.  Point E is on side AB such that ECB = 50°.  Find, with proof, the measure of EDB.

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Solution by Construction

Mark K on AC such that KBC = 20°.  Draw KB and KE.

BEC = ECB, and so BEC is isosceles with BE = BC.
BKC = BCK, and so BKC is isosceles with BK = BC.
Therefore BE = BK. EBK = 60°, and so EBK is equilateral.
BDK = DBK = 40° and so BDK is isosceles, with KD = KB = KE.
So KDE is isosceles, with EKD = 40°, since EKC = 140°.
Therefore EDK = 70°, yielding EDB = 30°.