The goal of this activity is to determine if we can conclude that two triangles are congruent with only some information about their sides and angles. [br][br]In order to show that they are congruent, we will attempt to map one onto the other, by [b][u]mapping each vertex onto its corresponding vertex, one at a time.[/u][/b]
Several segments and angles are measured in the triangles below. Does this diagram illustrate an example of SAS Congruence? Explain why / why not.
Although two sides and an angle of one triangle are congruent to two sides and an angle of another triangle, the [b][u]angle is not included between the two sides[/u][/b], so it is not SAS.
We will begin by making sure that vertex A corresponds with vertex J. Using the same diagram above, construct a vector to map A onto J then translate [math]\Delta[/math]ABE along that vector.
The figures, after the translation performed in step 2, are shown in the box below. Next, we will try to make a second set of vertices map onto one another. Which vertices are already coinciding? Why does this make sense?
Without performing any transformations, B' and H are coinciding. Since the congruent angles have been laid on top of each other (by the translation) and the lengths of AB and JH are the same, it follows that B and H would coincide.
Is there a transformation that will allow us to make E' coincide with I? Explain.
No - since two of the vertices already coincide, we are trying to make E' coincide with I using a translation, reflection or rotation. The length of JE' needs to increase in order for the third vertex to coincide, so it is not possible using those transformations.
We have shown that two triangles that have two pairs of congruent sides and a pair of (NON-INCLUDED) congruent angles cannot be mapped onto one another. What does this mean about two triangles that meet the SSA criteria?
Triangles that meet the SSA criteria are not congruent.