Given: Triangle ABC  Triangle DEF and <C  <F Solve for x and find the measure of the acute angles in the right triangles. What triangle theorems does this solution use? A˚A˚ (4x – 8 ) ˚ Warm Up B˚B˚ C˚C˚ D˚D˚ E˚E˚ F˚F˚ (x – 7 ) ˚

Proving Theorems about Triangles Theorems are true statements that follows as a result of other true statements A two-column proof has numbered statements and reasons that show the logical order of an argument A paragraph proof is a proof that has the same information as a two-column proof; but is written in a paragraph

Properties of Congruent Segments and Triangles Reflexive Property AB  AB and  ABC   ABC Symmetric Property If AB  CD, then CD  AB  ABC   DEF then  DEF   ABC Transitive Property If AB  CD and CD  EF, then AB  EF If  ABC   DEF, and  DEF   JKL, then  ABC   JKL

Properties of Congruent Segments and Triangles Substitution Property If a = b than a can be substituted for b in an equation or expression If AB=CD, then AB can be substituted for CD

Lesson 2.2 Proving Triangles Congruent Example 1State the Property [a] [b] A B C DE F A B C D E

Example 2 ATriangle Proofs Given: See Diagram Prove: A B C D Statements Reasons E 1 2

Example 2 BTriangle Proofs Given: ABCD is a Rectangle Prove: Statements Reasons A B C D Triangle Proofs Part I Worksheet

Warm-Up (2.2) Given: StatementsReasons1. Prove: A B C D E Given VA = AIA = AAS Triangle Proof Review Worksheet

Lesson 2.3 Proving Triangles Congruent & CPCTC Example 1State Properties [a] [b] U W X Y Z AB CD

Example 2 ACPCTC Given: See Diagram Prove: A B C D Statements Reasons E Given VA = SAS CPCTC

Example 2 B Given: Prove: Statements Reasons A B C D Triangle Proofs Part II Worksheet Given AIA = Reflex. SAS CPCTC

Warm-UpWarm-Up Use the following order pairs: A(2, 4) and B(-2, -6) [1]Find the slope AB [2]Find the slope // and | to AB [3]Find the length of AB (simplify the radical)

Math ISkill Review Solving Basic Quadratic Equations Step for Solving (Factoring Method) [1]Set equation equal to zero [2]Factor the non-zero side [3]Identify the zeros of each factor (zero product property – take the opposite value) Examples Worksheet 2.6

Example 1 Factoring Method [A]m 2 – m – 6 = 0 (m + 3)(m – 2) = 0 {–3, 2} [B]m 2 – 9m + 20 = 0 (m – 4)(m – 5) = 0 {4, 5}

Example 1 Factoring Method [C]x 2 + 5x – 36 = 0 (x + 4)(x – 9) = 0 {– 4, 9} [D]n n + 45 = 0 (n + 3)(n + 15) = 0 {–3, –5}

Example 1 Factoring Method [E]x 2 = 12x – 20 (x – 2)(x – 10) = 0 {2, 10} [F]n 2 – 100 = 48n (n + 2)(n – 50) = 0 {–2, 50} x 2 – 12x + 20 = 0 n 2 – 48x – 100 = 0 Examples Worksheet 2.6