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Convex Quadrilateral And Intersection (Posted on 2007-10-03) Difficulty: 3 of 5
PQRS is a convex quadrilateral with diagonals PR and QS that intersect at the point T. It is known that, Angle QPR = 50o, Angle RPS = 60o, Angle RQS = 30o and Angle QSR = 25o

Determine Angle PTQ.

See The Solution Submitted by K Sengupta    
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Solution Harder than it looks | Comment 1 of 8
At first it looked easy, then it looked impossible, then I got it.

Call Angle PTQ by a.  All other angles can be easily written in terms of a.
WLOG let PS=1 to set a scale

[1] by Law of Sines on Triangle PQS, RS=sin(155-a)/sin(60)
[2] by Law of Sines on Triangle PTS, TS=sin(60)/sin(a)
[3] Using TS from [2] and Law of Sines on TRS, RS=sin(60)/sin(155-a)

[1] and [3] show that RS is its own reciprocal, so RS=1
Therefore Triangle PRS is isosceles.
[4] 60 = 155-a
[5] a = 95

  Posted by Jer on 2007-10-03 10:53:09
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