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What is Congruence?
In geometry, congruence refers to the exact same size and shape.
If two figures are congruent, you could perfectly overlap them if you moved, rotated, or flipped one of them.
Key Concepts
Corresponding Parts: When two figures are congruent, their corresponding sides and angles have the same measure.
Corresponding Sides: Sides that match up when the figures are placed on top of each other.
Corresponding Angles: Angles that match up when the figures are placed on top of each other.
Congruence Symbols:
The symbol for congruence is "≅".
For example, if triangle ABC is congruent to triangle DEF, we write: ΔABC ≅ ΔDEF
Tests for Congruence (for Triangles):
SSS (Side-Side-Side): If all three sides of one triangle are congruent to the corresponding sides of another triangle, the triangles are congruent.
SAS (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to the corresponding sides and included angle of another triangle, the triangles are congruent.
ASA (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to the corresponding angles and included side of another triangle, the triangles areAAS (Angle-Angle-Side):** If two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, the triangles are congruent.
HL (Hypotenuse-Leg): (For right triangles only) If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, the triangles are congruent.
LL (Leg-Leg): If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.
Applications
Geometry proofs: Congruence theorems are used to prove that two triangles are congruent, which can then be used to prove other geometric relationships.
Real-world applications: Congruence is important in many fields, such as engineering, architecture, and manufacturing, where precise measurements and shapes are crucial.