Proving Δs are  : SSS, SAS, HL, ASA, & AAS

Objectives Use the SSS Postulate Use the SAS Postulate Use the HL Theorem Use ASA Postulate Use AAS Theorem CPCTC Theorem

Vocabulary Bisect: to cut into two equal parts

Postulate (SSS) Side-Side-Side  Postulate If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are .

More on the SSS Postulate If seg AB  seg ED, seg AC  seg EF, & seg BC  seg DF, then ΔABC  ΔEDF. E D F A B C

EXAMPLE 1 Use the SSS Congruence Postulate Write a proof. GIVEN KL NL, KM NM PROVE KLM NLM Proof It is given that KL NL and KM NM By the Reflexive Property, LM LN. So, by the SSS Congruence Postulate, KLM NLM

Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent. GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. SOLUTION Yes. The statement is true. DFG HJK Side DG HK, Side DF JH,and Side FG JK. So by the SSS Congruence postulate, DFG HJK.

GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. 2. ACB CAD SOLUTION BC AD GIVEN : PROVE : ACB CAD PROOF: It is given that BC AD By Reflexive property AC AC, But AB is not congruent CD.

GUIDED PRACTICE for Example 1 Therefore the given statement is false and ABC is not Congruent to CAD because corresponding sides are not congruent

GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. QPT RST 3. SOLUTION QT TR , PQ SR, PT TS GIVEN : PROVE : QPT RST PROOF: It is given that QT TR, PQ SR, PT TS. So by SSS congruence postulate, QPT RST. Yes the statement is true.

Postulate (SAS) Side-Angle-Side  Postulate If 2 sides and the included  of one Δ are  to 2 sides and the included  of another Δ, then the 2 Δs are .

More on the SAS Postulate If seg BC  seg YX, seg AC  seg ZX, & C  X, then ΔABC  ΔZXY. B Y ) ( A C X Z

EXAMPLE 2 Use the SAS Congruence Postulate Write a proof. GIVEN BC DA, BC AD PROVE ABC CDA STATEMENTS REASONS Given BC DA S Given BC AD BCA DAC Alternate Interior Angles Theorem A AC CA Reflexive Property of Congruence S

EXAMPLE 2 Use the SAS Congruence Postulate STATEMENTS REASONS ABC CDA SAS Congruence Postulate

Given: RS  RQ and ST  QT Prove: Δ QRT  Δ SRT. Example 3: Given: RS  RQ and ST  QT Prove: Δ QRT  Δ SRT. S Q R T

R Q R Example 3: T Statements Reasons________ 1. RS  RQ; ST  QT 1. Given 2. RT  RT 2. Reflexive 3. Δ QRT  Δ SRT 3. SSS Postulate

Given: DR  AG and AR  GR Prove: Δ DRA  Δ DRG. Example 4: Given: DR  AG and AR  GR Prove: Δ DRA  Δ DRG. D R A G

Example 4: Statements_______ 1. DR  AG; AR  GR 2. DR  DR 3.DRG & DRA are rt. s 4.DRG   DRA 5. Δ DRG  Δ DRA Reasons____________ 1. Given 2. Reflexive Property 3.  lines form 4 rt. s 4. Right s Theorem 5. SAS Postulate D R G A

Theroem (HL) Hypotenuse - Leg  Theorem If the hypotenuse and a leg of a right Δ are  to the hypotenuse and a leg of a second Δ, then the 2 Δs are . Note: Right Triangles Only

Postulate (ASA): Angle-Side-Angle Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.

Theorem (AAS): Angle-Angle-Side Congruence Theorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.

Proof of the Angle-Angle-Side (AAS) Congruence Theorem Given: A  D, C  F, BC  EF Prove: ∆ABC  ∆DEF D A B F C Paragraph Proof You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF. E

Theroem (CPCTC) Corresponding Parts of Congruent Triangles are Congruent When two triangles are congruent, there are 6 facts that are true about the triangles: the triangles have 3 sets of congruent (of equal length) sides and  the triangles have 3 sets of congruent (of equal measure) angles. Use this after you have shown that two figures are congruent. Then you could say that Corresponding parts of the two congruent figures are also congruent to each other.

Example 5: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Example 5: In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. Thus, you can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

Example 6: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Example 6: In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

Example 7: Given: AD║EC, BD  BC Prove: ∆ABD  ∆EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC. Use the fact that AD ║EC to identify a pair of congruent angles.

Writing Proofs Proofs are used to prove what you are finding. Geometric proofs can be written in one of two ways: two columns, or a paragraph. A paragraph proof is only a two-column proof written in sentences List the given statements and then list the conclusion to be proved Draw a figure and mark the figure accordingly along with your proofs

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given If || lines, then alt. int. s are  Vertical Angles Theorem ASA Congruence Postulate