4.4 Isosceles Triangles, Corollaries, & CPCTC

1. 4.4 Isosceles Triangles,Corollaries, &CPCTC

2. Isosceles Triangles • Has at least 2 congruent sides. • The angles opposite the congruent sides are congruent • Converse is also true.The sides opposite the congruent angles are also congruent. • This is a COROLLARY. A corollary naturally follows a theorem or postulate. We can prove it if we need to, but it really makes a lot of sense.

3. Vertex angle Base • The bisector of the vertex angle of an isoscelesΔ is the perpendicular bisector of the base. In addition, you just learned that the angles opposite congruent sides are congruent…

4. B F That means that EG  CB A E What is AC congruent to? FE G C Corresponding parts When you use a shortcut (SSS, AAS, SAS, ASA, HL) to show that 2 triangles are , that means that ALL the corresponding parts are congruent. EX: If a triangle is congruent by ASA (for instance), then all the other corresponding parts are .

5. Corresponding parts of congruent triangles are congruent. Corresponding parts of congruent triangles are congruent. Corresponding parts of congruent triangles are congruent. Corresponding parts of congruent triangles are congruent.

6. Corresponding Parts of Congruent Triangles are Congruent. If you can prove congruence using a shortcut, then you KNOW that the remaining corresponding parts are congruent. CPCTC You can only use CPCTC in a proof AFTER you have proved congruence.

7. Statements Reasons AC  DF Given C  F Given CB  FE Given ΔABC  ΔDEF SAS AB  DE CPCTC For example: A Prove: AB  DE B C D F E

8. Using CPCTC • History According to legend, • one of Napoleon’s officers used • congruent triangles to estimate • the width of a river. On the • riverbank, the officer stood up • straight and lowered the visor • of his cap until the farthest thing • he could see was the edge of the • opposite bank. He then turned • and noted the spot on his side • of the river that was in line with • his eye and the tip of his visor