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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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