Triangle Congruence Exploration

Author:
Ben Graber
If we know that two shapes are congruent, we know that the corresponding parts are congruent and can apply the measurements of one shape to the other. What are the fewest measurements needed to prove two triangles are congruent? Use the applet below to help determine what measurements are necessary to prove congruence. SSS = side-side-side (when you have three side lengths) SAS = side-angle-side (when you have two side lengths and the angle BETWEEN them) SSA = side-side-angle (when you have two side lengths and an angle NOT between them) ASA = angle-side-angle (when you have two angles and the side length between them) AAS = angle-angle-side (when you have two angles and a side length that is not between them) AA = angle-angle (when you have two angle measurements)
NOTE: two of the six postulates do NOT prove congruence! a. Create two triangles with side lengths of 3, 4, and 5 units. Are these triangles congruent? Is it possible to create two triangles with the same side lengths that are not congruent? Is the SSS postulate enough to prove that triangles are congruent? b. Create two triangles with side lengths of 3 and 3 units with a 40 degree angle BETWEEN the 3 unit sides. Are these triangles congruent? Is it possible to create two triangles with the same side-angle-side configuration that are not congruent? Is the SAS postulate enough to prove that triangles are congruent? c. Create two triangles with side lengths of 3 and 1.5 units with a 25 degree angle that is NOT between the two given sides. Is it possible to create two triangles with the same side-side-angle configuration that are not congruent? Is the SSA postulate enough to prove that triangles are congruent? d. Create two triangles with angle measurements of 45 and 45 degrees with a 5 unit length side BETWEEN the 45 degree angles. Is it possible to create two triangles with the same angle-side-angle configuration that are not congruent? Is the ASA postulate enough to prove that triangles are congruent? e. Create two triangles with angle measurements of 30 and 90 degrees with a 4 unit side that is NOT between the two given angles. Is it possible to create two triangles with the same angle-angle-side configuration that are not congruent? Is the AAS postulate enough to prove that triangles are congruent? f. Create two triangles with angle measurements of 60, 40, and 80 degrees. Is it possible to create two triangles with the same angle-angle-angle configuration that are not congruent? Is the AA or AAA postulate enough to prove that triangles are congruent?