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Determine if two triangles are congruent using SSS, SAS, ASA, AAS, or HL postulates
Two triangles are congruent if they have the same shape and size. This means all corresponding sides and angles are equal. There are five main methods to prove triangle congruence:
If all three sides of one triangle are equal to the corresponding sides of another triangle, the triangles are congruent.
If two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
If two angles and the included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
If two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
For right triangles only: If the hypotenuse and one leg are equal, the triangles are congruent.
Congruent means identical in shape and size. Congruent triangles have all corresponding sides and angles equal, though they may be positioned differently.
SSA (Side-Side-Angle) is not a valid congruence postulate because two sides and a non-included angle can create two different triangles (ambiguous case), except in the special case of right triangles where it becomes HL.
ASA requires the side to be between the two angles (included side), while AAS has the side adjacent to one angle but not between both angles (non-included side). Both prove congruence.
HL (Hypotenuse-Leg) only applies to right triangles. If both triangles have a 90-degree angle and their hypotenuses and one leg are equal, they are congruent.
No, AAA (Angle-Angle-Angle) only proves similarity, not congruence. Triangles with the same angles can have different sizes, so they're similar but not necessarily congruent.
CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent." Once you prove triangles are congruent, you know all their corresponding parts are equal.
Corresponding parts are matched based on their position. The smallest side corresponds to the smallest side, the largest angle to the largest angle, etc. Proper labeling helps identify these relationships.
Congruence transformations (translations, rotations, and reflections) move or flip a triangle without changing its size or shape. If one triangle can be transformed into another, they are congruent.
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