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Geometric reasoning

  1. Mar 24, 2016 #1
    1. The problem statement, all variables and given/known data
    As shown in the diagram (attached), ABDF is a rhombus, ACE is an equilateral triangle, and AB  AC . Find ABC through geometric reasoning (a scale diagram will gain no credit).

    2. See drawing (picture attached) Screen Shot 2016-03-24 at 21.36.19.png


    3. The attempt at a solution
    If I make angle ABC = x then BCA is also = x (as triangle ABC is isosceles) so I got x + x + 180 - 2x = 180

    angles ACE = CEA = EAC = 60 degrees (Equilateral triangle)

    Must I use alternate angles to help me solve this problem...
     
  2. jcsd
  3. Mar 24, 2016 #2

    haruspex

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    You need to find some way to use that ABDF is a rhombus, not just a parallelogram.
     
  4. Mar 24, 2016 #3
    if BCA = x then CAF = x (alternate angles)

    CAE = 60 but what is EAF = ?
     
  5. Mar 24, 2016 #4

    haruspex

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    What has that to do with my hint? What is the difference between a rhombus and a general parallelogram?
     
  6. Mar 24, 2016 #5
    sides are equal
     
  7. Mar 24, 2016 #6
    how does that help me?
     
  8. Mar 24, 2016 #7
    Is BC a 1/3 of BD?
     
  9. Mar 24, 2016 #8

    haruspex

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    Maybe more isosceles triangles?
     
  10. Mar 24, 2016 #9
    So we have ABC = BDF = DFA = x
     
  11. Mar 24, 2016 #10

    haruspex

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    No, BDF is not equal to the other two.
    You need to use that it is a rhombus, not just a parallelogram. Find two sides that are equal in the rhombus but might not be in a parallelogram.
     
  12. Mar 24, 2016 #11
    So ABC = DFA = x

    in each triangle we get x + x + 180 - 2x = 180 (but then this is obvious)
     
  13. Mar 24, 2016 #12

    haruspex

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    I repeat:
     
  14. Mar 24, 2016 #13
    In quadrilateral CEFA we have 60+x+x+180-2x+60+60=360 but then 360 = 360 leads nowhere
     
  15. Mar 24, 2016 #14
    all 4 sides are equal in a rhombus
     
  16. Mar 24, 2016 #15

    haruspex

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    Yes, but some are also equal in a parallelogram. I repeat
     
  17. Mar 24, 2016 #16
    i give up
     
  18. Mar 24, 2016 #17

    haruspex

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    As you said, all four sides are equal in a rhombus. That is not true of a parallelogram. In a parallelogram, which sides must be equal? Which sides need not be equal? Answer in terms of 'adjacent' and 'opposite'.
     
  19. Mar 24, 2016 #18
    Opposite
     
  20. Mar 24, 2016 #19

    haruspex

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    Which question is that the answer to? I asked which must be equal and which need not be equal. The answers are different, obviously.
     
  21. Mar 24, 2016 #20
    As you said, all four sides are equal in a rhombus. That is not true of a parallelogram. In a parallelogram, which sides must be equal? Which sides need not be equal? Answer in terms of 'adjacent' and 'opposite'.

    In the rhombus ABDF, AB = BD = DF = FA
    In a parallelogram opposite = equal, adjacent = not equal
     
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