4.3 to 4.5 Proving Δ s are : SSS, SAS, HL, ASA, & AAS

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4.3 to 4.5 Proving Δ s 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. Postulate 19 ( SSS ) Side-Side-Side  Postulate. - PowerPoint PPT Presentation

Transcript of 4.3 to 4.5 Proving Δ s are : SSS, SAS, HL, ASA, & AAS

  • ObjectivesUse the SSS Postulate Use the SAS PostulateUse the HL TheoremUse ASA PostulateUse AAS Theorem

  • Postulate 19 (SSS)Side-Side-Side PostulateIf 3 sides of one are to 3 sides of another , then the s are .

  • More on the SSS PostulateIf seg AB seg ED, seg AC seg EF, & seg BC seg DF, then ABC EDF.

  • EXAMPLE 1Use the SSS Congruence Postulate

  • GUIDED PRACTICEfor Example 1Decide whether the congruence statement is true. Explain your reasoning.SOLUTION

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

  • GUIDED PRACTICEfor Example 1Decide whether the congruence statement is true. Explain your reasoning.SOLUTION

  • Postulate 20 (SAS)Side-Angle-Side Postulate If 2 sides and the included of one are to 2 sides and the included of another , then the 2 s are .

  • More on the SAS PostulateIf seg BC seg YX, seg AC seg ZX, & C X, then ABC ZXY.BACXYZ)(

  • EXAMPLE 2Use the SAS Congruence PostulateWrite a proof.GIVENPROVE

  • EXAMPLE 2Use the SAS Congruence PostulateSTATEMENTSREASONS ABC CDASAS Congruence Postulate

  • Given: RS RQ and ST QT Prove: QRT SRT.QRSTExample 3:

  • Statements Reasons________ 1. RS RQ; ST QT 1. Given 2. RT RT 2. Reflexive 3. QRT SRT 3. SSS PostulateExample 3:RQRT

  • Given: DR AG and AR GRProve: DRA DRG. DARGExample 4:

  • Statements_______1. DR AG; AR GR2. DR DR3.DRG & DRA are rt. s4.DRG DRA5. DRG DRAReasons____________1. Given 2. Reflexive Property3. lines form 4 rt. s

    4. Right s Theorem 5. SAS Postulate

    DAGRExample 4:

  • Theroem 4.5 (HL)Hypotenuse - Leg TheoremIf the hypotenuse and a leg of a right are to the hypotenuse and a leg of a second , then the 2 s are .

  • Postulate 21(ASA):Angle-Side-Angle Congruence PostulateIf 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 4.6 (AAS): Angle-Angle-Side Congruence TheoremIf 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 TheoremGiven: A D, C F, BC EFProve: ABC DEFParagraph 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.ABCDEF

  • 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: ADEC, BD BCProve: ABD EBCPlan 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.

  • Proof:Statements:BD BCAD ECD C ABD EBC ABD EBCReasons:GivenGivenIf || lines, then alt. int. s are Vertical Angles TheoremASA Congruence Postulate

  • AssignmentGeometry: Workbook pg 67 - 75

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