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 $\displaystyle \Delta ABC$ and $\displaystyle \Delta ADC$ with: .$\displaystyle AB = AD,\;BC = CD$

. . and, of course, $\displaystyle AC = AC.$

Since $\displaystyle AB = AD,\;\Delta ABD$ is isosceles.

Hence, its base angles are equal: .$\displaystyle \angle ABD = \angle ADB = \alpha$

Since $\displaystyle BC = CD,\;\Delta BCD$ is isosceles.

Hence, its base angles are equal: .$\displaystyle \angle CBD = \angle CDB = \beta$

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

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

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