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**HOMEWORK: WS - Congruent Triangles**

Are They Congruent? Proving Δ’s are using: SSS, SAS, HL, ASA, & AAS HOMEWORK: WS - Congruent Triangles

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**Methods of Proving Triangles Congruent**

SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. SAS If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. ASA If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. AAS If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. HL If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent. Methods of Proving Triangles Congruent

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**Example: OR DIRECT Information Direct information comes in two forms:**

congruent statements in the ‘GIVEN:’ part of a proof marked in the picture Example: GIVEN KL NL, KM NM OR PROVE KLM NLM

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**Example: INDIRECT Information**

Indirect Information appears in the ‘GIVEN:’ part of the proof but is NOT a congruency statement Example: J Given: JO SH; O is the midpoint of SH Prove: SOJ HOJ S O H

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INDIRECT Information Perpendicular lines right angles all rt ∠s are ≅ Midpoint of a segment 2 ≅ segments Parallel lines AIA Parallelogram 2 sets of parallel lines 2 pairs of AIA Segment is an angle bisector 2 ≅ angles Segments bisect each other 2 sets of ≅ segments Perpendicular bisector of a segment 2 ≅ segments & 2 right angles

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**Built- in information is part of the drawing.**

Example: Vertical angles VA Shared side Reflexive Property Shared angle Reflexive Property Any Parallelogram 2 pairs parallel lines 2 pairs of AIA

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**Steps to Write a Proof Take the 1st Given and MARK it on the picture**

WRITE this Given in the PROOF & its reason If the Given is NOT a ≅ statement, write the ≅ stmt to match the marks Continue until there are no more GIVEN 4. Do you have 3 ≅ statements? If not, look for BUILT-IN parts 5. Do you have ≅ triangles? If not, write CNBD If YES, Write the triangle congruency and reason (SSS, SAS, SAA, ASA, HL)

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**ΔKLM ≅ ΔNLM SSS GIVEN KL NL, KM NM ≅ ≅ ≅ PROVE KLM NLM 𝐾𝐿 𝑁𝐿 given**

𝐿𝑀 𝐿𝑀 reflexive prop ΔKLM ≅ ΔNLM SSS

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**ΔABC ≅ ΔCDA SAS GIVEN ≅ BC DA, BC AD PROVE ΔABC ≅ ΔCDA BC DA ≅ given**

∠BCA ∠DAC ≅ AIA AC AC ≅ reflexive prop ΔABC ≅ ΔCDA SAS

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**∆ABC ∆DEF AAS Given: A D, C F, 𝐵𝐶 𝐸𝐹 Prove: ∆ABC ∆DEF D**

A D given C E C F given 𝐵𝐶 𝐸𝐹 given ∆ABC ∆DEF AAS

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Given: 𝐿𝐽 bisects IJK, ILJ JLK Prove: ΔILJ ΔKLJ 𝐿𝐽 bisects IJK Given IJL IJH Definition of angle bisector ILJ JLK Given 𝐽𝐿 𝐽𝐿 Reflexive Prop ΔILJ ΔKLJ ASA

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** ΔTUV ΔWXV SAS Given: 𝑇𝑊 ≅ 𝑉𝑊 , 𝑈𝑉 ≅ 𝑉𝑋 Prove: ΔTUV ΔWXV**

TVU WVX Vertical angles ΔTUV ΔWXV SAS

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** ΔHIJ ΔLKJ ASA Given: 𝐻𝐽 ≅ 𝐽𝐿 , H L Prove: ΔHIJ ΔLKJ**

H L Given IJH KJL Vertical angles ΔHIJ ΔLKJ ASA

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**ΔPRT ΔSTR SAS Given: 𝑃𝑅 ≅ 𝑆𝑇 , PRT STR Prove: ΔPRT ΔSTR**

PRT STR Given 𝑅𝑇 ≅ 𝑅𝑇 Reflexive Prop ΔPRT ΔSTR SAS

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**ΔABM ≅ ΔPBM SAS Given: 𝑀𝐵 is perpendicular bisector of 𝐴𝑃**

Prove: ∆𝐴𝐵𝑀≅ ∆𝑃𝐵𝑀 𝑀𝐵 is perpendicular bisector of 𝐴𝑃 given ∠ABM & ∠PBM are rt ∠s def lines ∠ABM ≅ ∠PBM all rt ∠s are ≅ 𝐴𝐵 ≅ 𝐵𝑃 def bisector 𝐵𝑀 ≅ 𝐵𝑀 reflexive prop. ΔABM ≅ ΔPBM SAS

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**ΔMON ≅ ΔQOP SAS Given: O is the midpoint of 𝑀𝑄 and 𝑁𝑃**

Prove: ΔMON ≅ ΔPOQ O is the midpoint of 𝑀𝑄 and 𝑁𝑃 given 𝑀𝑂 ≅ 𝑂𝑄 def. midpoint 𝑁𝑂 ≅ 𝑂𝑃 def. midpoint ∠MON ≅ ∠QOP VA ΔMON ≅ ΔQOP SAS

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**ΔABD ≅ ΔCDB SAS Given: 𝐴𝐷 ≅ 𝐶𝐷 ; 𝐴𝐷 || 𝐶𝐷 Prove: ΔABD ≅ ΔCDB**

∠ADB ≅ ∠CBD AIA 𝐵𝐷 ≅ 𝐵𝐷 reflexive prop. ΔABD ≅ ΔCDB SAS

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**Given: 𝐽𝑂 𝑆𝐻 ; O is the midpoint of 𝑆𝐻 Prove: SOJ HOJ**

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**Given: HJ GI, GJ JI Prove: ΔGHJ ΔIHJ**

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**Given: 1 2; A E ; C is midpt of AE Prove: ΔABC ΔEDC**

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**ΔPQR ΔPSR HL Given: 𝑃𝑄 𝑄𝑅 , 𝑃𝑆 𝑆𝑅 , and 𝑄𝑅 𝑆𝑅**

Prove: ΔPQR ΔPSR 𝑃𝑄 𝑄𝑅 Given PQR = 90° Def. lines 𝑃𝑆 𝑆𝑅 Given PSR = 90° Def. lines PQR PSR all right s are 𝑄𝑅 𝑆𝑅 Given 𝑃𝑅 𝑃𝑅 Reflexive Prop ΔPQR ΔPSR HL

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Checkpoint Decide if enough information is given to prove the triangles are congruent. If so, state the congruence postulate you would use.

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**Given: LJ bisects IJK, ILJ JLK Prove: ΔILJ ΔKLJ**

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**ΔABC ΔEDC ASA Given: 1 2, A E and 𝐴𝐶 𝐸𝐶**

Prove: ΔABC ΔEDC 1 2 Given A E Given 𝐴𝐶 𝐸𝐶 Given ΔABC ΔEDC ASA

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Given: 𝐴𝐵 𝐶𝐵 , 𝐴𝐷 𝐶𝐷 Prove: ΔABD ΔCBD 𝐴𝐵 𝐶𝐵 Given 𝐴𝐷 𝐶𝐷 Given 𝐵𝐷 𝐵𝐷 Reflexive Prop ΔABD ΔCBD SSS

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