Triangle Congruence Students will be able to apply the Triangle Congruence Postulates in order to solve problems.

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Proving Triangles Congruent

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Triangle Congruence Students will be able to apply the Triangle Congruence Postulates in order to solve problems.

Unit F2 Triangle Rigidity The property of triangle rigidity gives you a shortcut for proving two triangles are congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. Because of this property, do we need to know all six facts (that all pairs of sides and all pairs of angles are congruent) to show that two triangles are congruent?

Unit F3 Do we need all six facts? These are the five postulates or theorems that prove that two triangles are congruent using less that all six facts. NO! SSS SAS ASA AAS HL

Unit F4 Side-Side-Side Congruence (SSS) If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. If,, and then A B C D E F

Unit F5 –∠ A is the included angle between and. –∠ B is the included angle between and. –∠ C is the included angle between and. Included Angles An included angle is an angle formed by two adjacent sides of a polygon. A C B

Unit F6 Side-Angle-Side Congruence (SAS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. If,, and then A B C D E F ΔABC ≅ ΔDEF

Unit F7 – is the included side between ∠ A and ∠ B. – is the included side between ∠ B and ∠ C. – is the included side between ∠ A and ∠ C. Included Sides An included side is the common side of two consecutive angles in a polygon. A C B

Unit F8 Angle-Side-Angle Congruence (ASA) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. If,, and then A B C D E F ΔABC ≅ ΔDEF

Unit F9 Angle-Angle-Side Congruence (AAS) If two angles and a nonincluded side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent. If,, and then A B C D E F ΔABC ≅ ΔDEF

Unit F10 Hypotenuse-Leg Congruence (HL) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent. If and then A B C D E F ΔABC ≅ ΔDEF

Unit F11 Are there other Postulates? These are the only five Postulates or Theorems that prove that two triangles are congruent using less that all six facts. SSS SAS ASA AAS HL Warning! There are no Angle-Angle-Angle or Side-Side-Angle Postulates of Congruence!

Unit F12 Are there other Postulates? This is an example of why there is no Angle- Angle-Angle Postulate of Congruence. These two triangles have all three pairs of angles congruent, but the two triangles are not congruent.

Unit F13 Are there other Postulates? This is an example of why there is no Side-Side- Angle Postulate of Congruence. These two triangles have two pairs of sides and a pair of nonincluded angles congruent, but the two triangles are not congruent.

Unit F14 Name the Postulate or Theorem

FHSUnit F15 Lesson Quiz Which postulate, if any, can be used to prove the triangles congruent?