In geometry, the term "congruence" refers to the idea that two shapes are identical in shape and size, even if they are positioned differently in space. Theorems regarding congruence are essential in proving properties about triangles, polygons, and other geometric shapes.
Here are the primary congruence theorems for triangles:
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Side-Side-Side (SSS) Congruence Theorem: If three sides of one triangle are equal in length to three sides of another triangle, then the two triangles are congruent.
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Side-Angle-Side (SAS) Congruence Theorem: If two sides and the included angle of one triangle are equal in length to two sides and the included angle of another triangle, then the two triangles are congruent.
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Angle-Side-Angle (ASA) Congruence Theorem: If two angles and the side between them of one triangle are equal to two angles and the side between them of another triangle, then the two triangles are congruent.
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Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a side not between them in one triangle are equal to two angles and the corresponding side in another triangle, then the two triangles are congruent.
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Hypotenuse-Leg (HL) Congruence Theorem: This is specific to right triangles. If the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
These theorems can be used to establish that two triangles are congruent, which can help prove further properties or relationships in geometric figures.