Two shapes are congruent if all of their corresponding parts (angles and sides) are congruent.
Parts correspond by both being smallest, medium, or largest in their respective shapes.
We learned that there are some shortcuts to showing whether or not two triangles are congruent. Triangles have 6 parts (3 sides, 3 angles). We learned that there are certain starting structures that can only be completed one way:
SSS- if all 3 side lengths are given or set, there is only one way to complete the triangle
SAS- if 2 side lengths and the angle between them are given or set, there is only one way to complete the triangle
AAS- if two angles and a side connected to one of those angles are given or set, then there is only one way to complete the triangle.
ASA- if two triangles have two angles and the side connecting them given or set, then there is only one way to complete the triangle.
The idea of “there is only one way to complete the triangle” is crucial: If two things start off in a way that can only be finished in one way, the two things must end the same way. For triangles, if two of them start with the same 3 side lengths, all of the angles are going to end up being exactly the same (congruent) too! Since there is only one way to put 3 given side lengths together, the two triangles with the same 3 side lengths must be congruent.
When writing a triangle congruence statement, make sure to take a moment to make sure the letter order of the second triangle represents the angles and sides that are actually corresponding and congruent.
Ex: Triangle ABC is congruent to Triangle DFE means that AB=DF, BC=FE, AC=DE, <A=<D, <B=<F, and <C=<E (Please note, I would use the actual congruence symbol for these statements, but it is not available on the capabilities for this webpage)
Reflexive Property of Congruence
All things are congruent to themselves. (Ex: AB=AB, <A=<A)
We use the reflexive property for when two triangles have a shared or common side. This side is the same in both triangles, so we say it is congruent to itself.
Also be on the lookout for Vertical Angles and angle pairs formed by transversals of parallel lines: They will allow you to identify angles that are congruent to build a case for two triangles actually being congruent.
The last proof strategy we learned about was CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Go ahead and read that last statement in parenthesis again. It should make sense. CPCTC is the abbreviation for that idea. When we use SAS, SSS, ASA, or AAS to show two triangles are congruent, we all of a sudden get to identify all of the other corresponding parts as congruent as well. I think of it as “buy 3, get 3 free.” You know 3 sides are congruent, so you end up knowing the 3 angles are congruent as well. This is CPCTC
Below is an example of how this all comes together in proofs:
The above proof is the last thing we did, to begin talking about Isosceles triangles.
Recall: Isosceles triangles are triangles with 2 congruent sides.
The third side that isn’t congruent to the other 2 is called the base. The angles connected to the base are called the base angles. The base angles of an isosceles triangle are always across from the two congruent sides. As you can see above, the base angles theorem says that the base angles of an isosceles triangle are congruent.