## Tuesday, August 15, 2023

### Geometry Problem 1551: Unraveling Angle Relations in Cyclic Quadrilaterals: Solving for Angle GEJ

Challenging Geometry Puzzle: Problem 1551. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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#### 3 comments:

1. < FBE = < FAE = < CAD = < FGE = a (say)
< JHE = < HEJ = < BDH = < BDA = b (say)
< FEB = < FAB = < FGB = u (say)
< HEC = < HJC = < HDC = v (say)
< ABG = < AFG = < AEG = t (say)
< JCD = < JHD = < JED = y (say)

a + b = 52.........(1) (Triangle BCE)
u + v + 52 + 46 = 180 (angles on Point E)
So u +v = 82 .......(2)

u + 2a + t = 180 - 52 = 128...(3) (Angles on Point G)
v + 2b + y = 180 - 52 = 128 ...(4) (Angles on Point J)

(3)+(4) ==> u + v + 2a + 2b + t +y = 256

So 82 + 104 + t + y = 256
t + y = 70

< GEJ = 180 - (a+t) - (b+y) = 180 - (a+b) - (t+y) = 180 - 52 - 70
< GEJ = 58

Sumith Peiris
Moratuwa
Sri Lanka

2. Because < BDC= < BAC;
So, < HFE= < FHE
Same for <EGJ = EJG
So < FHE =90-46/2 = 67 degree
So < EDC = 67 degree
So < ECD =180 -52-67 = 61 degree
So < EJG =61 degree
So <GEJ = 2*(90-61)= 58 degree

3. Antonio - a new problem arises from this problem

Prove that FHJG is a cyclic quadrilateral