# Two Sides and an Angle?

## What If You Have SSS?

In this construction, two of the pairs of corresponding sides are automatically congruent (do you see why?) Adjust the vertices of the triangle on the left to make the third pair of corresponding sides also congruent. Is it possible to form two NON-congruent triangles here?

## What If You Have SAS?

In this construction, two pairs of corresponding sides are again automatically congruent. This time, adjust the vertices of the triangle on the left so that the INCLUDED angle is congruent to the corresponding angle in Triangle ABC. Is it possible to form two NON-congruent triangles here?

## What If You Have SSA?

In this construction, two pairs of corresponding sides are yet again automatically congruent. However, one of the adjacent angles is also forced to be congruent to the corresponding angle in Triangle ABC. Move the vertices of the "triangle" on the left to actually form a triangle. Is it possible to form two NON-congruent triangles?

## The Special Case of Right Triangles: HL

When we are comparing two right triangles, we automatically have a pair of congruent corresponding angles (the right angles), so if the pair of hypotenuses and one pair of legs are also congruent, we have two sides and an angle. Is this a special case of SAS or SSA?
• Go back to the appropriate exploration and make Triangle ABC a right triangle (with its corresponding angle in the other triangle also a right angle). Is it possible to form two NON-congruent right triangles here?
Thinking about this another way, if we know HL, the Pythagorean Theorem tells us the other pair of legs must also be congruent, so we also have SSS. Is what you found out about HL above consistent with what you found out about SSS?

## Mapping One Congruent Triangle Onto Another

To make sure you understand the "recipe" (see Page 2) for mapping one congruent triangle onto another, you might want to try mapping Triangle ABC to a congruent partner in each of the GeoGebra task windows above. Or to challenge yourself further, can you accomplish the mapping in fewer transformations than when following the recipe?

## Up Next...

Now that we have considered the various potential 3-letter criteria involving two sides and an angle, we'll next consider two angles and a side.