Triangle Congruence
Notation
To denote equal sides two polygons, we mark corresponding sides with a dash. If multiple sides are equal, different corresponding sides will have a different number of dashes.
To denote equal angles of two polygons, we mark corresponding angles with an arch through the angle. If multiple angles are equal, different corresponding angles will have a different number of arches or a different number of dashes through a single arch.
The notation for congruence is an equals sign "=" with a similarity symbol on top.
Determining Triangle Congruence
Triangles that are congruent have the exact same side lengths and angle measurements. The sides and angles do not have to be in the same position, however; congruent triangles can be translated (shifted), reflected (flipped), or rotated versions of each other.
There are five (5) methods to determine whether two triangles are congruent. These are side-side-side (SSS), side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), and hypotenuse-leg (HL).
1) Side-side-side (SSS)
The side-side-side (SSS) method states that if all three sides of two triangles are congruent, the triangles are congruent.
2) Side-angle-side (SAS)
The side-angle-side (SAS) method states that if two sides and the angle formed by the two sides (included angle) are congruent, the triangles are congruent.
3) Angle-side-angle (ASA)
The angle-side-angle (ASA) method states that if two angles and the side in between (included side) are congruent, the triangles are congruent.
4) Angle-angle-side (AAS)
The angle-angle-side (AAS) method states that if two angles and a non-included side are congruent, the triangles are congruent.
5) Hypotenuse-leg (HL)
The hypotenuse-leg (HL) method only applies to right triangles. It states that if the hypotenuses and one leg are congruent, the triangles are congruent.
Common Mistakes to Avoid
1) Side-side-angle (SSA)
If two sides and a non-included angle are congruent, the triangles are not necessarily congruent. This is because you can swing the side opposite the fixed angle and not change its length.
2) Angle-angle-angle (AAA)
If all three angles of both triangles are the same, the two triangles are not necessarily congruent. Three congruent corresponding angles only shows that the triangles are similar.
