Lesson 4.4 Triangle Congruencies

Two geometric figures with exactly the same size and shape. The Idea of a Congruence A C B DE F

How much do you need to know... need to know about two triangles to prove that they are congruent?

Corresponding parts of congruent triangles Triangles that are the same size and shape are congruent triangles. Each triangle has three angles and three sides. If all six corresponding parts are congruent, then the triangles are congruent.

Corresponding parts of congruent triangles A C B X Z Y If ΔABC is congruent to ΔXYZ, then vertices of the two triangles correspond in the same order as the letter naming the triangles. ΔABC = ΔXYZ ~

Corresponding parts of congruent triangles A C B X Z Y This correspondence of vertices can be used to name the corresponding congruent sides and angles of the two triangles. ΔABC = ΔXYZ ~

If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. Corresponding Parts  ABC   DEF B A C E D F 1.AB  DE 2.BC  EF 3.AC  DF 4.  A   D 5.  B   E 6.  C   F

∠ A ≅ ∠ P ∠ B ≅ ∠ Q ∠ C ≅ ∠ R Corresponding Angles Corresponding Sides ∆ABC ≅ ∆PQR

Do we need to use all six pairs to prove two triangles are congruent? Proving Two Triangles are Congruent

Do you need all six ? NO ! WHY?

That’s why… That’s why there are some shortcuts (but applicable to TRIANGLES ONLY). These shortcuts, if used correctly, will help you prove triangle congruency. Remember that congruency means EXACT size and shape… don’t confuse it with “similar”.

Shortcuts in Triangle Congruency We learned that two triangles were congruent if EACH corresponding pair of angles were congruent AND each pair of corresponding sides were congruent. Remember that you had to LIST EACH PAIR? Yikes! That could get very long and tedious (tiring).

Properties of Triangle Congruence Congruence of triangles is reflexive, symmetric, and transitive.REFLEXIVE K J L K J L ΔJKL = ΔJKL ~

Properties of Triangle Congruence Congruence of triangles is reflexive, symmetric, and transitive.SYMMETRIC K J L Q P R If ΔJKL = ΔPQR, then ΔPQR = ΔJKL. ~ ~

Properties of Triangle Congruence Congruence of triangles is reflexive, symmetric, and transitive.TRANSITIVE K J L Q P R If ΔJKL = ΔPQR, and ΔPQR = ΔXYZ, then ΔPQR = ΔXYZ, then ΔJKL = ΔXYZ. ΔJKL = ΔXYZ. ~ ~ ~ Y X Z