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Proving Δ’s are using: SSS, SAS, HL, ASA, & AAS HOMEWORK: WS - Congruent Triangles Methods of Proving Triangles Congruent SSS SAS ASA AAS HL If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent. DIRECT Information Direct information comes in two forms: congruent statements in the ‘GIVEN:’ part of a proof marked in the picture Example: GIVEN PROVE KL NL, KM KLM NM NLM OR 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 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 BUILT-IN Information 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 Steps to Write a Proof 1. Take the 1st Given and MARK it on the picture 2. WRITE this Given in the PROOF & its reason 3. 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) GIVEN KL ≅ NL, KM ≅ NM PROVE 𝐾𝐿 𝑁𝐿 KLM ≅ NLM given 𝐾𝑀 𝑁𝑀 given 𝐿𝑀 𝐿𝑀 reflexive prop ΔKLM ≅ ΔNLM SSS GIVEN BC ≅ DA, BC AD PROVE ΔABC ≅ ΔCDA BC ≅ DA BC AD given given ∠BCA ≅ ∠DAC AC ≅ AC AIA reflexive prop ΔABC ≅ ΔCDA SAS Given: A D, C F, 𝐵𝐶 𝐸𝐹 Prove: ∆ABC ∆DEF A D B F A D given C F given 𝐵𝐶 𝐸𝐹 given ∆ABC ∆DEF C E AAS Given: 𝐿𝐽 bisects IJK, ILJ JLK Prove: ΔILJ ΔKLJ 𝐿𝐽 bisects IJK IJL IJH ILJ JLK 𝐽𝐿 𝐽𝐿 I J L K Given Definition of angle bisector Given Reflexive Prop ΔILJ ΔKLJ ASA Given: 𝑇𝑊 ≅ 𝑉𝑊, 𝑈𝑉 ≅ 𝑉𝑋 Prove: ΔTUV ΔWXV U W 𝑇𝑊 ≅ 𝑉𝑊 𝑈𝑉 ≅ 𝑉𝑋 TVU WVX Given T Given Vertical angles ΔTUV ΔWXV SAS V X Given: 𝐻𝐽 ≅ 𝐽𝐿, H L Prove: ΔHIJ ΔLKJ I 𝐻𝐽 ≅ 𝐽𝐿 H L IJH KJL K Given Given Vertical angles J H ΔHIJ ΔLKJ ASA L Given: 𝑃𝑅 ≅ 𝑆𝑇, PRT STR Prove: ΔPRT ΔSTR 𝑃𝑅 ≅ 𝑆𝑇 PRT STR 𝑅𝑇 ≅ 𝑅𝑇 R S Given Given Reflexive Prop ΔPRT ΔSTR SAS P T 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 𝐵𝑀 ≅ 𝐵𝑀 def bisector reflexive prop. ΔABM ≅ ΔPBM 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 Given: 𝐴𝐷 ≅ 𝐶𝐷; 𝐴𝐷 || 𝐶𝐷 Prove: ΔABD ≅ ΔCDB 𝐴𝐷 ≅ 𝐶𝐷 given 𝐴𝐷 || 𝐶𝐷 given ∠ADB ≅ ∠CBD AIA 𝐵𝐷 ≅ 𝐵𝐷 reflexive prop. ΔABD ≅ ΔCDB SAS J Given: 𝐽𝑂 𝑆𝐻; O is the midpoint of 𝑆𝐻 Prove: SOJ HOJ S 0 H Given: HJ GI, GJ JI Prove: ΔGHJ ΔIHJ H G J I Given: 1 2; A E ; C is midpt of AE Prove: ΔABC ΔEDC B D 1 A 2 C E Given: 𝑃𝑄 𝑄𝑅 , 𝑃𝑆 𝑆𝑅 , and 𝑄𝑅 𝑆𝑅 Prove: ΔPQR ΔPSR 𝑃𝑄 𝑄𝑅 PQR = 90° 𝑃𝑆 𝑆𝑅 PSR = 90° PQR PSR 𝑄𝑅 𝑆𝑅 𝑃𝑅 𝑃𝑅 Given P Def. lines Given Def. lines all right s are Given Reflexive Prop ΔPQR ΔPSR HL Q R S Checkpoint Decide if enough information is given to prove the triangles are congruent. If so, state the congruence postulate you would use. Given: LJ bisects IJK, ILJ JLK Prove: ΔILJ ΔKLJ I J L K Given: 1 2, A E and 𝐴𝐶 𝐸𝐶 B Prove: ΔABC ΔEDC 1 2 A E 𝐴𝐶 𝐸𝐶 Given Given Given ΔABC ΔEDC D 1 A ASA 2 C E Given: 𝐴𝐵 𝐶𝐵 , 𝐴𝐷 𝐶𝐷 Prove: ΔABD ΔCBD 𝐴𝐵 𝐶𝐵 𝐴𝐷 𝐶𝐷 𝐵𝐷 𝐵𝐷 A Given Given Reflexive Prop ΔABD ΔCBD C B SSS D