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Congruence in Triangles

Congruence in Triangles. Information. Proving triangles congruent. Triangles are congruent when corresponding angles and corresponding sides are congruent. A triangle has three sides and three angles, so there are six relationships that must be true for two triangles to be congruent.

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Congruence in Triangles

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  1. Congruence in Triangles

  2. Information

  3. Proving triangles congruent Triangles are congruent when corresponding angles and corresponding sides are congruent. A triangle has three sides and three angles, so there are six relationships that must be true for two triangles to be congruent. AE and BD bisect each other. Prove that △ABC≅△DEC. A given from the figure:    ∠A≅ ∠E ∠B≅ ∠D AB≅ DE D need to show: C B  vertical angle theorem ∠ACB≅ ∠DCE  BC≅ CD a bisector divides a segment into two congruent segments E  AC≅ CE

  4. Triangle rigidity Triangle rigiditygives a shortcut for proving two triangles congruent. It means that if the side lengths of a triangle are given, the triangle can have only one shape. How many triangles can you construct with side of lengths 3cm, 4cm and 5cm? 3 The triangles may appear different, but they are congruent. 5 4 4 5 5 3 4 If the sides of a triangle are fixed, so are the angles. 3

  5. Rigidity in other polygons How many quadrilaterals can you construct with four congruent sides? Infinitely many different quadrilaterals can have four congruent sides. Different quadrilaterals can have congruent sides, but have different angles. Unlike triangles, the sides do not determine the angles. This is because quadrilaterals are not rigid. In fact, triangles are the only rigid polygon.

  6. Triangle congruence postulates Because triangles are rigid, if the side lengths are fixed, the triangle can have only one shape. This means that to prove that two triangles are congruent, you only need to show that corresponding sides are congruent. This is the side-side-side congruence postulate. There are several similar postulates involving angles that can be used to prove triangles congruent, such as: • side-angle-side congruence postulate • angle-side-angle congruence postulate • angle-angle-side congruence postulate.

  7. Side-side-angle? Side-side-angle (SSA) cannot be used to prove triangles congruent. Find an example to illustrate this. △ZYX △WYX Y ZY≅ WY side: 76° YX≅ YX 22° side: angle: ∠X≅ ∠X 128° 52° 30° X Z  but△ZYX ≇ △WYX W because,∠YZW ≇ ∠YWX. Even though two sides and a nonincluded angle are congruent, the triangles are not congruent.

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