Developing Congruence Criteria (ASA)

STEP 0

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]

STEP 1

Measure segments AC and DF.[br]Measure angles A, C, D and F.[br][br]What conclusion(s) can you draw? Write them below.

STEP 2

We will begin by making sure that vertex A corresponds with vertex D. Using the same diagram above, construct a vector to map A onto D then translate [math]\Delta[/math]ABC along that vector.

STEP 3

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. What transformation will make C' map exactly onto F? How do you know that C'' and F will coincide?

STEP 4

Perform the transformation described in Step 3 in the box below. Be sure that C'' and F coincide exactly.

STEP 5

The figures, after the rotation performed in step 5, are shown in the box below. Two vertices of the triangle already coincide. Describe the transformation that will make B'' coincide with E. How do you know that they will coincide?

STEP 6

Perform the transformation described in Step 5 in the box below (note: you can only reflect a figure over a line, not a segment. First, construct a line over the segment and then perform the reflection). Be sure that B''' and E coincide exactly.

STEP 7

We have shown that two triangles that have two pairs of congruent angles and a pair of congruent included sides can be mapped onto one another. So from this point forward, when we see a diagram like the one below, what congruence statement can be made?