What have we done so far with triangles?

Talked about what it meant for two shapes to be congruent- for all of their pairs of corresponding parts to be congruent.

We can show this by writing congruence statements: ∆ABC is congruent to ∆DFE (the order of the letters matters, it shows what parts of the two triangles correspond).

Investigated how to create a triangle using rulers, protractors and compasses.

Given all 3 sides and all 3 angles of a triangle, we tried to determine if it matters what is connected to what- it turned out that the smallest angle needed to be across from the smallest side, same with medium and large sides and angles.

First learned the idea of “SAS” structure (Given a Side, Angle, and Side connected at the angle)- only one way to complete the triangle. (Try it, show it to yourself!).

If a structure can only be completed one way, and two triangles start *with the same parts* in the same structure, then the two triangles must both *end* the same way (They are congruent).

We then investigated the other variations of being given *structures* of *three* Angles and Sides. If a structure can only be completed in one way, then it can be used to show two triangles are congruent. If it has more than one way to be completed, then even if two triangles start with the same corresponding parts in the same structure, you can’t conclude they will both be completed in the same way.

Triangle Congruence Structures: SSS, SAS, ASA, AAS

If two triangles have the same (congruent) corresponding parts, in the same structure, then they are congruent.

Structures that Don’t Work: AA(A), SSA

AA(A) can be completed infinitely many ways- inconclusive

SSA can be completed 2 ways- inconclusive

CPCTC- Corresponding Parts of Congruent Triangles are Congruent

Once you know two triangles are congruent, you can then conclude that their other corresponding parts will also be congruent. We only need three parts in a congruence structure to conclude two triangles are congruent, but *then* we also know about the other three parts by CPCTC.

We also worked on classifying triangles by their angles and sides using the words:

Equilateral, Isosceles, or Scalene and Right, Obtuse, Acute, or Equiangular.

We learned about various theorems that have to do with triangles and their angles:

Triangle Sum Theorem, Exterior Angle Theorem, Base Angles Theorem, Converse of the Base Angles Theorem, Third Angles Theorem.

We applied them to solve equations and find unknown values on triangles.