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Proof concerning similar triangles and their ratios

  1. Sep 13, 2015 #1
    1. The problem statement, all variables and given/known data
    Prove that similar triangles have equal ratios (ratios of the sides)

    2. Relevant equations

    3. The attempt at a solution
    I posted a rather messy and incorrect proof and problem statement prior to this and I wish to correct my mistakes now.
    The ratios of the sides are sin(α), cos(α) and tan(α). The ratios of the similar triangle are sin(α'),cos(α') and tan(α').
    So the problem amounts to showing that if two triangles are similar sin(α)=sin(α'), cos(α)=cos(α') and tan(α)=tan(α').
    A triangle is similar to another if one or more of the above rules(SSS,AAA...) apply, so I shall attempt to prove that the ratios are the same for each case. I begin with the easiest:

    SSS: All ratios are the same. Nothing to prove.
    AAA: All angles are the same. It follows that α=α'→sin(α)=sin(α')...
    SAS: If one angle and the enclosing sides are the same two triangles are similar.
    I must now prove that all side ratios are the same. If the angle that coincides with that of the similar triangle is not the right angle, for instance α, then:
    180°-90°-α=β so all angles are the same and I have already proven the AAA case. Job done hopefully
    SSA: If the ratio of two sides and the angle opposite to the greater side of the ratio coincide then a triangle is similar to another. I must again prove that all ratios are then equivalent.
    There are all in all three ratios(sin,cos,tan), so I shall prove this for all three seperatly:

    Proof for sinus: All that is known is that the sini of the triangles are the same, I must also prove that cosinus and tangens are also the same. Mathematically:

    sin(α)=sin(α') since sin(90°-α')=sin(90°-α)=cos(α)→cos(α')=cos(α) ( Please tell me if this is infact true)
    and tan(α)=sin(α)/cos(α)=sin(α')/cos(α')=tan(α')

    I am concerned about the last proof, I would be very greatfull if you could correct it.
    Thank you.
  2. jcsd
  3. Sep 13, 2015 #2


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    What definition of similar triangles are you using? THe definition I'm familiar with is AAA, so with that definition you don't need to do all the other stuff. THe other rules are for congruent triangles, not just similar.
    Last edited: Sep 13, 2015
  4. Sep 13, 2015 #3


    Staff: Mentor

    In your other thread I showed that SSA doesn't guarantee that two triangles are congruent, and even included a rough drawing of two triangles that are obviously not congruent. Please revisit your other thread to see my post.
  5. Sep 13, 2015 #4


    Staff: Mentor

    I agree. @moriheru, what definition of "similar triangles" are you using?
    Last edited: Sep 13, 2015
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