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Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Lesson 26: Triangle Congruency Proofs—Part I Student Outcomes Students complete proofs requiring a synthesis of the skills learned in the last four lessons. Classwork 1. Given: ̅̅̅̅, 𝑩𝑪 ̅̅̅̅ ⊥ 𝑫𝑪 ̅̅̅̅. ̅̅̅̅ ⊥ 𝑩𝑪 𝑨𝑩 ̅̅̅̅̅ 𝑫𝑩 bisects ∠𝑨𝑩𝑪, ̅̅̅̅ 𝑨𝑪 bisects ∠𝑫𝑪𝑩. ̅̅̅̅. ̅̅̅̅ ⋍ 𝑬𝑪 𝑬𝑩 Prove: △ 𝑩𝑬𝑨 ≅△ 𝑪𝑬𝑫. ̅̅̅̅, 𝑩𝑪 ̅̅̅̅ ⊥ 𝑫𝑪 ̅̅̅̅ ̅̅̅̅ ⊥ 𝑩𝑪 𝑨𝑩 Given 𝒎∠𝑨𝑩𝑪 = 𝟗𝟎˚, 𝒎∠𝑫𝑪𝑩 = 𝟗𝟎˚ ̅̅̅̅ bisects ∠𝑫𝑪𝑩 ̅̅̅̅̅ bisects ∠𝑨𝑩𝑪, 𝑨𝑪 𝑫𝑩 Def. of perpendicular 𝒎∠𝑨𝑩𝑬 = 𝟒𝟓˚, 𝒎∠𝑫𝑪𝑬 = 𝟒𝟓˚ Def. of bisect 𝒎∠𝑨𝑩𝑬 = 𝒎∠𝑫𝑪𝑬 Transitive Property of = (Substitution) ∠𝑨𝑩𝑬 ⋍ ∠𝑫𝑪𝑬 ̅̅̅̅ 𝑬𝑩 ⋍ ̅̅̅̅ 𝑬𝑪. Def. of congruent angles ∠𝑨𝑬𝑩 𝒂𝒏𝒅 ∠𝑫𝑬𝑪 are vertical angles Definition of Vert. ∠𝒔 ∠𝑨𝑬𝑩 ⋍ ∠𝑫𝑬𝑪 Vert. ∠ Theorem △ 𝑩𝑬𝑨 ≅△ 𝑪𝑬𝑫 ASA Lesson 26: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Given Given Triangle Congruency Proofs—Part I 4/29/17 202 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 26 M1 GEOMETRY 2. Given: ̅̅̅̅ ⋍ 𝒀𝑲 ̅̅̅̅, 𝑷𝑿 ̅̅̅̅ ⋍ 𝑷𝒀 ̅̅̅̅, ∠𝒁𝑿𝑱 ⋍ ∠𝒁𝒀𝑲. 𝑿𝑱 ̅̅̅ = ̅̅̅̅̅ Prove: 𝑱𝒀 𝑲𝑿. ̅𝑿𝑱 ̅̅̅ ⋍ ̅̅̅̅ ̅̅̅̅ 𝒀𝑲, 𝑷𝑿 ⋍ ̅̅̅̅ 𝑷𝒀, ∠𝒁𝑿𝑱 ⋍ ∠𝒁𝒀𝑲. Given ∠𝑱𝒁𝑿 𝒂𝒏𝒅 ∠𝑲𝒁𝒀are vertical angles Definition of Vert. ∠𝒔 ∠𝑱𝒁𝑿 ⋍ ∠𝑲𝒁𝒀 Vert. ∠ 𝑻𝒉𝒆𝒐𝒓𝒆𝒎 △ 𝑱𝒁𝑿 ≅△ 𝑲𝒁𝒀 SAA (AAS) ∠𝑱 ⋍ ∠𝑲 Corresponding Parts of Congruent Triangles are Congruent (CPCTC) ∠𝑷 ⋍ ∠𝑷 ̅̅̅̅ ⋍ 𝑷𝑲 ̅̅̅̅̅ 𝑱𝑷 Reflexive Property △ 𝑷𝑱𝒀 ≅△ 𝑷𝑲𝑿 ̅̅̅ = ̅̅̅̅̅ 𝑱𝒀 𝑲𝑿 SAA When congruent segments are added to congruent segments, the sums are conguent Lesson 26: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org CPCTC Triangle Congruency Proofs—Part I 4/29/17 202 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 3. ̅̅̅̅ ̅̅̅̅. 𝑱𝑲 ⋍ ̅̅̅ 𝑱𝑳, ̅̅̅̅ 𝑱𝑲 ∥ 𝑿𝒀 ̅̅̅̅ ⋍ 𝑿𝑳 ̅̅̅̅. Prove: 𝑿𝒀 ̅̅̅̅ 𝑱𝑲 ⋍ ̅̅̅ 𝑱𝑳 Given: Given ∠𝑲 ⋍ ∠𝑳 ̅̅̅̅ ̅̅̅̅ 𝑱𝑲 ∥ 𝑿𝒀 Base ∠𝒔 in isos. △are congruent ∠𝑲 ⋍ ∠𝒀 Corr. ∠𝒔 are congruent since ̅̅̅̅ 𝑱𝑲 ∥ ̅̅̅̅ 𝑿𝒀 ∠𝒀 ⋍ ∠𝑳 ̅̅̅̅ ⋍ 𝑿𝑳 ̅̅̅̅ 𝑿𝒀 Substitution/Transitive Given Base ∠𝒔 converse Lesson 26: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Triangle Congruency Proofs—Part I 4/29/17 202 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 26 M1 GEOMETRY 4. Given: Prove: ∠𝟏 ⋍ ∠𝟐, ∠𝟑 ⋍ ∠𝟒. ̅̅̅̅ ⋍ 𝑩𝑫 ̅̅̅̅̅. 𝑨𝑪 ∠𝟏 ⋍ ∠𝟐 ̅̅̅̅ 𝑩𝑬 ⋍ ̅̅̅̅ 𝑪𝑬 Given ∠𝟑 ⋍ ∠𝟒 Given ∠𝑨𝑬𝑩 𝒂𝒏𝒅 ∠𝑫𝑬𝑪 are vertical angles Definition of Vert. ∠𝒔 ∠𝑨𝑬𝑩 ⋍ ∠𝑫𝑬𝑪 Vert. ∠𝒔 Theorem △ 𝑨𝑩𝑪 ≅ △ 𝑫𝑪𝑩 ASA ∠𝑨 ⋍ ∠𝑫 ̅̅̅̅ 𝑩𝑪 ⋍ ̅̅̅̅ 𝑩𝑪 CPCTC △ 𝑨𝑩𝑪 ≅ △ 𝑫𝑪𝑩 ̅̅̅̅ ⋍ 𝑩𝑫 ̅̅̅̅̅ 𝑨𝑪 SAA Base ∠𝒔 converse Reflexive Property CPCTC OR ∠𝟏 ⋍ ∠𝟐 ̅̅̅̅ 𝑩𝑬 ⋍ ̅̅̅̅ 𝑪𝑬 Given ∠𝟑 ⋍ ∠𝟒 Given ∠𝑨𝑬𝑩 𝒂𝒏𝒅 ∠𝑫𝑬𝑪 are vertical angles Definition of Vert. ∠𝒔 ∠𝑨𝑬𝑩 ⋍ ∠𝑫𝑬𝑪 Vert. ∠𝒔 Theorem △ 𝑨𝑩𝑪 ≅ △ 𝑫𝑪𝑩 ̅̅̅̅ ⋍ 𝑬𝑫 ̅̅̅̅ 𝑨𝑬 ASA ̅̅̅̅ 𝑨𝑪 ⋍ ̅̅̅̅̅ 𝑩𝑫 If congruent segments are added to congruent segments, then their sums are congruent Base ∠𝒔 converse Lesson 26: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org CPCTC Triangle Congruency Proofs—Part I 4/29/17 202 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 5. Given: ̅̅̅̅ 𝑨𝑩 ≅ ̅̅̅̅ 𝑨𝑪, ̅̅̅̅ 𝑹𝑩 ≅ ̅̅̅̅ 𝑹𝑪, Prove: ̅̅̅̅ ≅ 𝑺𝑪 ̅̅̅̅. 𝑺𝑩 ̅̅̅̅ 𝑨𝑩 ≅ ̅̅̅̅ 𝑨𝑪, ̅̅̅̅ 𝑹𝑩 ≅ ̅̅̅̅ 𝑹𝑪 ̅̅̅̅ ≅ 𝑨𝑹 ̅̅̅̅ 𝑨𝑹 Given △ 𝑨𝑹𝑪 ≅△ 𝑨𝑹𝑩 SSS ∠𝑨𝑹𝑪 ≅ ∠𝑨𝑹𝑩 CPCTC ∠𝑨𝑹𝑪 𝒂𝒏𝒅 ∠𝑺𝑹𝑪 form a linear pair Definition of a linear pair Reflexive Property ∠𝑨𝑹𝑩 𝒂𝒏𝒅 ∠𝑺𝑹𝑩 form a linear pair ∠𝑨𝑹𝑪 𝒂𝒏𝒅 ∠𝑺𝑹𝑪 are supplementary Linear Pair Theorem ∠𝑨𝑹𝑩 𝒂𝒏𝒅 ∠𝑺𝑹𝑩 are supplementary ∠𝑺𝑹𝑪 ≅ ∠𝑺𝑹𝑩 ̅̅̅̅ 𝑺𝑹 = ̅̅̅̅ 𝑺𝑹 supplements of congruent angles are congruent △ 𝑺𝑹𝑩 ≅△ 𝑺𝑹𝑪 ̅̅̅̅ ≅ 𝑺𝑪 ̅̅̅̅ 𝑺𝑩 SAS Lesson 26: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Reflexive Property CPCTC Triangle Congruency Proofs—Part I 4/29/17 202 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 6. Given: Prove: ̅̅̅̅ ≅ 𝑱𝒀 ̅̅̅ 𝑱𝑿 ̅̅̅̅ ̅̅̅ ≅ ̅̅̅ 𝑱𝑲 ≅ ̅̅̅ 𝑱𝑳, ̅𝑱𝑿 𝑱𝒀. 𝑲𝑿 = 𝑳𝒀. Given ∠𝑱𝑿𝒀 ≅ ∠𝑱𝒀𝑿 Isosceles Triangle Theorem ∠𝑱𝑿𝑲 𝒂𝒏𝒅 ∠𝑱𝑿𝒀 form a linear pair Definition of Linear Pair ∠𝑱𝒀𝑳 𝒂𝒏𝒅 ∠𝑱𝒀𝑿 form a linear pair ∠𝑱𝑿𝑲 𝒂𝒏𝒅 ∠𝑱𝑿𝒀 are supplementary Linear Pair Theorem ∠𝑱𝒀𝑳 𝒂𝒏𝒅 ∠𝑱𝒀𝑿 are supplementary ∠𝑱𝑿𝑲 ≅ ∠𝑱𝒀𝑳 ̅̅̅̅ 𝑱𝑲 ≅ ̅̅̅ 𝑱𝑳 Supplements to congruent angles are congruent ∠𝑲 ≅ ∠𝑳 Isosceles Triangle Theorem △ 𝑱𝑿𝑲 ≅△ 𝑱𝒀𝑳 ̅̅̅̅̅ 𝑲𝑿 ≅ ̅̅̅̅ 𝑳𝒀 SAA Lesson 26: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Given CPCTC Triangle Congruency Proofs—Part I 4/29/17 202 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 7. Given: ̅̅̅̅ 𝑨𝑫 ⊥ ̅̅̅̅̅ 𝑫𝑹, ̅̅̅̅ 𝑨𝑩 ⊥ ̅̅̅̅ 𝑩𝑹, ̅̅̅̅ ≅ 𝑨𝑩 ̅̅̅̅. 𝑨𝑫 Prove: ∠𝑫𝑪𝑹 ≅ ∠𝑩𝑪𝑹. ̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅ 𝑨𝑫 ⊥ 𝑫𝑹, 𝑨𝑩 ⊥ ̅̅̅̅ 𝑩𝑹, Given ∠𝐀𝐃𝐑 𝐚𝐧𝐝 ∠𝐀𝐁𝐑 𝐚𝐫𝐞 𝐫𝐢𝐠𝐡𝐭 𝐚𝐧𝐠𝐥𝐞𝐬 Definition of perpendicular lines △ 𝐀𝐃𝐑 𝐚𝐧𝐝 △ 𝐀𝐁𝐑 𝐚𝐫𝐞 𝐫𝐢𝐠𝐡𝐭 𝐭𝐫𝐢𝐚𝐧𝐠𝐥𝐞𝐬 ̅̅̅̅ 𝑨𝑫 ≅ ̅̅̅̅ 𝑨𝑩 (Leg) Definition of right triangle ̅̅̅̅ ≅ 𝑨𝑹 ̅̅̅̅ (Hypotenuse) 𝑨𝑹 Reflexive Property △ 𝑨𝑫𝑹 ≅△ 𝑨𝑩𝑹 Hypotenuse Leg (HL) ∠𝑨𝑹𝑫 ≅ ∠𝑨𝑹𝑩 ∠𝑨𝑹𝑫 𝒂𝒏𝒅 ∠𝑫𝑹𝑪 𝒂𝒓𝒆 𝒂 𝑳𝒊𝒏𝒆𝒂𝒓 𝑷𝒂𝒊𝒓 Corr. ∠s of ≅△ (CPCFC/CPCTC) 𝑫𝒆𝒇𝒊𝒏𝒊𝒕𝒊𝒐𝒏 𝒐𝒇 𝑳𝒊𝒏𝒆𝒂𝒓 𝑷𝒂𝒊𝒓 Given ∠𝑨𝑹𝑩 𝒂𝒏𝒅 ∠𝑩𝑹𝑪 𝒂𝒓𝒆 𝒂 𝑳𝒊𝒏𝒆𝒂𝒓 𝑷𝒂𝒊𝒓 ∠𝑨𝑹𝑫 𝒂𝒏𝒅 ∠𝑫𝑹𝑪 𝒂𝒓𝒆 𝑺𝒖𝒑𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒂𝒓𝒚 𝑳𝒊𝒏𝒆𝒂𝒓 𝑷𝒂𝒊𝒓 𝑻𝒉𝒆𝒐𝒓𝒆𝒎 ∠𝑨𝑹𝑩 𝒂𝒏𝒅 ∠𝑩𝑹𝑪 𝒂𝒓𝒆 𝑺𝒖𝒑𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒂𝒓𝒚 ∠𝑫𝑹𝑪 ≅ ∠𝑩𝑹𝑪 ̅̅̅̅̅ 𝑫𝑹 ≅ ̅̅̅̅ 𝑩𝑹 Supplements of congruent angles are congruent ̅̅̅̅ ≅ 𝑹𝑪 ̅̅̅̅ 𝑹𝑪 Reflexive Property △ 𝑫𝑹𝑪 ≅△ 𝑩𝑹𝑪 SAS ∠𝑫𝑪𝑹 ≅ ∠𝑩𝑪𝑹 Corr. ∠s of ≅△ (CPCFC) Lesson 26: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Corr. sides of ≅△ (CPCFC) Triangle Congruency Proofs—Part I 4/29/17 202 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 8. Given: ̅̅̅̅, 𝑩𝑹 ̅̅̅̅, ̅̅̅̅ ≅ 𝑨𝑺 ̅̅̅̅ ≅ 𝑪𝑺 𝑨𝑹 ̅̅̅̅ ⊥ 𝑨𝑪 ̅̅̅̅. ̅̅̅̅ ⊥ 𝑨𝑩 ̅̅̅̅, 𝑺𝒀 𝑹𝑿 ̅̅̅̅ Prove: 𝑩𝑿 ≅ ̅̅̅̅ 𝑪𝒀. ̅̅̅̅ (S) ̅̅̅̅ ≅ 𝑨𝑺 𝑨𝑹 Given ∠𝑨𝑹𝑺 ≅ ∠𝑨𝑺𝑹 Base ∠s of isos. △ ≅ (𝑰𝑻𝑻) ∠𝑨𝑹𝑺 𝒂𝒏𝒅 ∠𝑨𝑹𝑩 are a Linear Pair 𝑫𝒆𝒇. 𝒐𝒇 𝒂 𝑳𝒊𝒏𝒆𝒂𝒓 𝑷𝒂𝒊𝒓 ∠𝑨𝑺𝑹 𝒂𝒏𝒅 ∠𝑨𝑺𝑪 𝒂𝒓𝒆 𝒂 𝑳𝒊𝒏𝒆𝒂𝒓 𝑷𝒂𝒊𝒓 ∠𝑨𝑹𝑺 𝒂𝒏𝒅 ∠𝑨𝑹𝑩 are Supplementary 𝑳𝒊𝒏𝒆𝒂𝒓 𝑷𝒂𝒊𝒓 𝑻𝒉𝒆𝒐𝒓𝒆𝒎 ∠𝑨𝑺𝑹 𝒂𝒏𝒅 ∠𝑨𝑺𝑪 𝒂𝒓𝒆 𝑺𝒖𝒑𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒂𝒓𝒚 ∠𝑨𝑹𝑩 ≅ ∠𝑨𝑺𝑪 (A) ̅̅̅̅ (S) ̅̅̅̅ ≅ 𝐂𝐒 𝐁𝐑 Supplements of congruent angles are congruent △ 𝑨𝑹𝑩 ≅△ 𝑨𝑺𝑪 SAS ∠𝑨𝑩𝑹 ≅ ∠𝑨𝑪𝑺 (A) ̅̅̅̅ ⊥ 𝐀𝐂 ̅̅̅̅ ̅̅̅̅ ⊥ 𝐀𝐁 ̅̅̅̅, 𝐒𝐘 𝐑𝐗 Corr. ∠s of ≅△ (CPCFC/CPCTC) ∠𝑹𝑿𝑩 𝒂𝒏𝒅 ∠𝑺𝒀𝑪 are right angles Def. of perpendicular ∠𝑹𝑿𝑩 ≅ ∠𝑺𝒀𝑪 (A) If two angles are right angles then they are congruent △ 𝑩𝑹𝑿 ≅△ 𝑺𝒀𝑪 ̅̅̅̅ ̅̅̅̅ ≅ 𝐂𝐘 𝐁𝐗 SAA Lesson 26: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Given Given Corr. sides of ≅△ (CPCFC) Triangle Congruency Proofs—Part I 4/29/17 202 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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