__Investigation#1__:

**1. **Draw a triangle on a sheet of paper and cut it out
with scissors.

2. Use it as a pattern to draw a second triangle and cut that triangle out.

Notice......If one triangle is placed on the other triangle, the two match exactly. This means that each part of the first triangle matches exactly the corresponding part of the second triangle.

You have just made a pair of congruent triangles.

** Congruent Triangles: Triangles that have the same size and shape.**

** Definition of Congruent
Triangles: **Two triangles are congruent if and
only if their corresponding parts are congruent. The symbol that
we use for congruent is @.

**Tests for Congruent Triangles**

Question: Is it always necessary to show that all of the corresponding parts (angles and sides) of two triangles are congruent to be sure that be sure that the two triangles are congruent?

** Investigation#2: **Construct a triangle with sides of length 3 inches,
5 inches, and 6 inches by following the steps below: (You will need a
straightedge and a compass)

1. On any line m, select a point A.

2. Construct segment AB on m such that AB=6 inches.

3. Using A as the center, draw an arc with radius 5 inches.

4. Using B as the center, draw an arc with radius 3 inches.

5. Let C be the point of intersection of the two arcs.

6. Draw segment AC and segment BC.

Try using the same procedure with AB=5 and then with AB=3, and compare all of the triangles. How many different triangles is it possible to construct with sides of the given measures?

** SSS Postulate:**
If the sides of one triangle are congruent to the sides of a second
triangle, then the triangles are congruent.

(The SSS Postulate can be used to prove triangles congruent.)

Click here for SSS Intermath description.

Will any other combinations of corresponding and congruent sides and angles determine a unique triangle?

Suppose you were given the measures of two sides and the angle that they form, which is called the included angle. How many different triangles would you be able to make?

By investigating this problem like we did with the SSS, you will find that the investigation will lead to the SAS postulate. Likewise with the ASA postulate.

__Additional
Postulates and Theorem that Prove Triangle Congruence__

** SAS Postulate: **If two sides and the included angle of one triangle
are congruent to two sides and the included angle of another triangle,
then the triangles are congruent.

Click here for SAS Intermath description.

** ASA Postulate: **If two angles and the included side of one triangle
are congruent to two angles and the included side of another triangle,
then the triangles are congruent.

Click here for ASA Intermath description.

** AAS Theorem**: If two angles and a nonincluded side
of one triangle are congruent to the corresponding two angles
and side of a second triangle, the two triangles are congruent.

Click here for AAS Intermath description.