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Math Help - Proving Side-Side-Side congruency of a triangle

  1. #1
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    Proving Side-Side-Side congruency of a triangle

    I need to prove SSS traingle congruency using Side-Angle-Side as an axiom. I know that if 3 sides of a triangle are congruent to 3 sides of another triangle the 2 triangles are equivalent. I am just not sure how to write up in a proof using SAS as the axiom or the given. Thanks for any help anyone can give. In the question it said to use a kite or dart to help with the proof, but I do not see how that can help.
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  2. #2
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    Hello, reagan3nc!

    I need to prove SSS triangle congruency using SAS as an axiom.

    It said to use a kite or dart to help with the proof
    Code:
                          B
                          *
                      * α |β*
                  *       |   *
              *           |     *
        A * - - - - - - - * - - - * C
              *           |E    *
                  *       |   *
                      * α |β*
                          *
                          D

    We are given two triangles with equal corresponding sides.
    Place them so they have an equal side in common.

    We have \Delta ABC and \Delta ADC with: . AB = AD,\;BC = CD
    . . and, of course, AC = AC.

    Since AB = AD,\;\Delta ABD is isosceles.
    Hence, its base angles are equal: . \angle ABD = \angle ADB = \alpha

    Since BC = CD,\;\Delta BCD is isosceles.
    Hence, its base angles are equal: . \angle CBD = \angle CDB = \beta


    Since \begin{Bmatrix}\angle ABC &=& \alpha + \beta \\\angle ACD &=& \alpha + \beta\end{Bmatrix} . then: . \angle ABC = \angle ADC


    And we have: . \begin{array}{ccc}\Delta ABC & &\Delta ADC \\ \hline AB &=& AD \\ \angle ABC &=& \angle ADC \\ BC &=& CD \end{array}


    Therefore: . \Delta ABC \cong \Delta ADC\;\text{ by SAS}

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