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No trigonometry, no algebra (Posted on 2005-12-23) Difficulty: 3 of 5
Look at the drawing below, where AB = BC = CD = DT = h, and TD perpendicular to AD:
                                   
                                   o T
                                   |
                                   | (h) 
                                   |
                                   |
  +----------+----------+----------o 
  A    (h)   B    (h)   C    (h)   D
The line AT makes an angle x with AD; the line BT makes an angle y with BD; the line CT makes an angle z with CD.

Using only geometry, prove that (angle x) + (angle y) = (angle z).

See The Solution Submitted by pcbouhid    
Rating: 3.0000 (1 votes)

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Solution Solution | Comment 6 of 10 |
 
   |CT|     h*sqrt(2)         h         |CB|
  ------ = ----------- = ----------- = ------
   |CA|        2*h        h*sqrt(2)     |CT|
Therefore, triangles ACT and TCB are similar.
Therefore, angles CAT and CTB are equal.
Angle z is an exterior angle of triangle CBT.
Therefore, angle(z) = angle(CBT) + angle(CTB)
                    = angle(CBT) + angle(CAT)
                    = angle(y) + angle(x)
 

  Posted by Bractals on 2005-12-24 04:35:23
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