In geometry, the study of triangles often involves analyzing relationships between sides and angles, as well as the rules that determine when two triangles are identical, or congruent.
Triangle Congruence Rules:
Two triangles are congruent if they have the exact same dimensions.
- SAS (Axiom 1): Two sides and the included angle are equal.
- ASA (Theorem 1): Two angles and the included side are equal.
- SSS (Theorem 4): All three sides are equal.
- RHS (Theorem 5): In right triangles, the hypotenuse and one side are equal.
Isosceles Triangle Properties
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Theorem 2: Angles opposite to equal sides are equal.
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Theorem 3: Sides opposite to equal angles are equal.
Inequalities in Triangles
When sides or angles are unequal, the following properties apply:
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Theorem 6: The angle opposite the longer side is larger.
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Theorem 7: The side opposite the larger angle is longer.
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Theorem 8: The sum of any two sides must be greater than the third side.