<h2><u>
These are my old notes, I hope they can help.</u></h2>
SSS, or Side Side Side
SAS, or Side Angle Side
ASA, or Angle Side Side
AAS, or Angle Angle Side
HL, or Hypotenuse Leg, for right triangles only
Side Side Side Postulate
A postulate is a statement taken to be true without proof. The SSS Postulate tells us,
If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
Congruence of sides is shown with little hatch marks, like this: ∥. For two triangles, sides may be marked with one, two, and three hatch marks.
If △ACE has sides identical in measure to the three sides of △HUM, then the two triangles are congruent by SSS:
Side Angle Side Postulate
The SAS Postulate tells us,
If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
△HUG and △LAB each have one angle measuring exactly 63°. Corresponding sides g and b are congruent. Sides h and l are congruent.
A side, an included angle, and a side on △HUG and on △LAB are congruent. So, by SAS, the two triangles are congruent.
Angle Side Angle Postulate
This postulate says,
If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
We have △MAC and △CHZ, with side m congruent to side c. ∠A is congruent to ∠H, while ∠C is congruent to ∠Z. By the ASA Postulate these two triangles are congruent.
Angle Angle Side Theorem
We are given two angles and the non-included side, the side opposite one of the angles. The Angle Angle Side Theorem says,
If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Here are congruent △POT and △LID, with two measured angles of 56° and 52°, and a non-included side of 13 centimeters:
[construct as described]
By the AAS Theorem, these two triangles are congruent.
HL Postulate
Exclusively for right triangles, the HL Postulate tells us,
Two right triangles that have a congruent hypotenuse and a corresponding congruent leg are congruent.
The hypotenuse of a right triangle is the longest side. The other two sides are legs. Either leg can be congruent between the two triangles.
Here are right triangles △COW and △PIG, with hypotenuses of sides w and i congruent. Legs o and g are also congruent:
[insert congruent right triangles left-facing △COW and right facing △PIG]
So, by the HL Postulate, these two triangles are congruent, even if they are facing in different directions.
Proof Using Congruence
Proving Congruent Triangles 5
Given: △MAG and △ICG
MC ≅ AI
AG ≅ GI
Prove: △MAG ≅ △ICG
Statement Reason
MC ≅ AI Given
AG ≅ GI
∠MGA ≅ ∠ IGC Vertical Angles are Congruent
△MAG ≅ △ICG Side Angle Side
If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
