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Lesson 25 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Lesson 25: Congruence Criteria for TrianglesโAAS and HL Student Outcomes ๏ง Students learn why any two triangles that satisfy the AAS or HL congruence criteria must be congruent. ๏ง Students learn why any two triangles that meet the AAA or SSA criteria are not necessarily congruent. Classwork Opening Exercise (7 minutes) Opening Exercise Write a proof for the following question. Once done, compare your proof with a neighborโs. Given: ๐ซ๐ฌ = ๐ซ๐ฎ, ๐ฌ๐ญ = ๐ฎ๐ญ ฬ ฬ ฬ ฬ is the angle bisector of โ ๐ฌ๐ซ๐ฎ Prove: ๐ซ๐ญ ๐ซ๐ฌ = ๐ซ๐ฎ Given ๐ฌ๐ญ = ๐ฎ๐ญ Given ๐ซ๐ญ = ๐ซ๐ญ Reflexive property โณ ๐ซ๐ฌ๐ญ โ โณ ๐ซ๐ฎ๐ญ SSS โ ๐ฌ๐ซ๐ญ โ โ ๐ฎ๐ซ๐ญ Corresponding angles of congruent triangles are congruent. ฬ ฬ ฬ ฬ ๐ซ๐ญ is the angle bisector of โ ๐ฌ๐ซ๐ฎ Definition of an angle bisector Exploratory Challenge (25 minutes) The included proofs of AAS and HL are not transformational; rather, they follow from ASA and SSS, already proved. Exploratory Challenge Today we are going to examine three possible triangle congruence criteria, Angle-Angle-Side (AAS), Side-Side-Angle (SSA), and Angle-Angle-Angle (AAA). Ultimately, only one of the three possible criteria will ensure congruence. Angle-Angle-Side Triangle Congruence Criteria (AAS): Given two triangles ๐จ๐ฉ๐ช and ๐จโฒ๐ฉโฒ๐ชโฒ. If ๐จ๐ฉ = ๐จโฒ๐ฉโฒ (Side), ๐ฆโ ๐ฉ = ๐ฆโ ๐ฉโฒ (Angle), and ๐ฆโ ๐ช = ๐ฆโ ๐ชโฒ (Angle), then the triangles are congruent. Proof: Consider a pair of triangles that meet the AAS criteria. If you knew that two angles of one triangle corresponded to and were equal in measure to two angles of the other triangle, what conclusions can you draw about the third angles of each triangle? Lesson 25: Date: Congruence Criteria for TrianglesโAAS and HL 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 204 Lesson 25 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Since the first two angles are equal in measure, the third angles must also be equal in measure. Given this conclusion, which formerly learned triangle congruence criteria can we use to determine if the pair of triangles are congruent? ASA Therefore, the AAS criterion is actually an extension of the ASA triangle congruence criterion. Note that when using the Angle-Angle-Side triangle congruence criteria as a reason in a proof, you need only state the congruence and โAAS.โ Hypotenuse-Leg Triangle Congruence Criteria (HL): Given two right triangles ๐จ๐ฉ๐ช and ๐จโฒ๐ฉโฒ๐ชโฒ with right angles ๐ฉ and ๐ฉโฒ, if ๐จ๐ฉ = ๐จโฒ๐ฉโฒ (Leg) and ๐จ๐ช = ๐จโฒ๐ชโฒ (Hypotenuse), then the triangles are congruent. Proof: As with some of our other proofs, we will not start at the very beginning, but imagine that a congruence exists so that triangles have been brought together such that ๐จ = ๐จโฒ and ๐ช = ๐ชโฒ; the hypotenuse acts as a common side to the transformed triangles. Similar to the proof for SSS, we add a construction and draw ฬ ฬ ฬ ฬ ฬ ๐ฉ๐ฉโฒ. โณ ๐จ๐ฉ๐ฉโฒ is isosceles by definition, and we can conclude that base angles ๐ฆโ ๐จ๐ฉ๐ฉโฒ = ๐ฆโ ๐จ๐ฉโฒ๐ฉ. Since โ ๐ช๐ฉ๐ฉโฒ and โ ๐ช๐ฉโฒ ๐ฉ are both the complements of equal angle measures (โ ๐จ๐ฉ๐ฉโฒ and โ ๐จ๐ฉโฒ๐ฉ), they too are equal in measure. Furthermore, since ๐ฆโ ๐ช๐ฉ๐ฉโฒ = ๐ฆโ ๐ช๐ฉโฒ ๐ฉ, the sides of โณ ๐ช๐ฉ๐ฉโฒ opposite them are equal in measure: ๐ฉ๐ช = ๐ฉโฒ๐ชโฒ. Then, by SSS, we can conclude โณ ๐จ๐ฉ๐ช โ โณ ๐จโฒ๐ฉโฒ๐ชโฒ. Note that when using the Hypotenuse-Leg triangle congruence criteria as a reason in a proof, you need only state the congruence and โHL.โ Lesson 25: Date: Congruence Criteria for TrianglesโAAS and HL 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 205 Lesson 25 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Criteria that do not determine two triangles as congruent: SSA and AAA Side-Side-Angle (SSA): Observe the diagrams below. Each triangle has a set of adjacent sides of measures ๐๐ and ๐, as well as the non-included angle of ๐๐°. Yet, the triangles are not congruent. Examine the composite made of both triangles. The sides of lengths ๐ each have been dashed to show their possible locations. The triangles that satisfy the conditions of SSA cannot guarantee congruence criteria. In other words, two triangles under SSA criteria may or may not be congruent; therefore, we cannot categorize SSA as congruence criterion. Angle-Angle-Angle (AAA): A correspondence exists between โณ ๐จ๐ฉ๐ช and โณ ๐ซ๐ฌ๐ญ. Trace โณ ๐จ๐ฉ๐ช onto patty paper, and line up corresponding vertices. Based on your observations, why isnโt AAA categorized as congruence criteria? Is there any situation in which ๐จ๐จ๐จ does guarantee congruence? Even though the angle measures may be the same, the sides can be proportionally larger; you can have similar triangles in addition to a congruent triangle. List all the triangle congruence criteria here: SSS, SAS, ASA, AAS, HL List the criteria that do not determine congruence here: SSA, AAA Lesson 25: Date: Congruence Criteria for TrianglesโAAS and HL 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 206 Lesson 25 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Examples (8 minutes) Examples 1. 2. Given: ฬ ฬ ฬ ฬ โฅ ๐ช๐ซ ฬ ฬ ฬ ฬ , ๐จ๐ฉ ฬ ฬ ฬ ฬ โฅ ๐จ๐ซ ฬ ฬ ฬ ฬ , ๐ฆโ ๐ = ๐ฆโ ๐ ๐ฉ๐ช Prove: โณ ๐ฉ๐ช๐ซ โ โณ ๐ฉ๐จ๐ซ ๐ฆโ ๐ = ๐ฆโ ๐ Given ฬ ฬ ฬ ฬ ๐จ๐ฉ โฅ ฬ ฬ ฬ ฬ ๐จ๐ซ Given ฬ ฬ ฬ ฬ ๐ฉ๐ช โฅ ฬ ฬ ฬ ฬ ๐ช๐ซ Given ๐ฉ๐ซ = ๐ฉ๐ซ Reflexive property ๐ฆโ ๐ + ๐ฆโ ๐ช๐ซ๐ฉ = ๐๐๐° Linear pairs form supplementary angles ๐ฆโ ๐ + ๐ฆโ ๐จ๐ซ๐ฉ = ๐๐๐° Linear pairs form supplementary angles ๐ฆโ ๐ช๐ซ๐ฉ = ๐ฆโ ๐จ๐ซ๐ฉ If two angles are equal in measure, then their supplements are equal in measure ๐ฆโ ๐ฉ๐ช๐ซ = ๐ฆโ ๐ฉ๐จ๐ซ = ๐๐° Definition of perpendicular line segments โณ ๐ฉ๐ช๐ซ โ โณ ๐ฉ๐จ๐ซ AAS Given: ฬ ฬ ฬ ฬ ๐จ๐ซ โฅ ฬ ฬ ฬ ฬ ฬ ๐ฉ๐ซ, ฬ ฬ ฬ ฬ ฬ ๐ฉ๐ซ โฅ ฬ ฬ ฬ ฬ ๐ฉ๐ช, ๐จ๐ฉ = ๐ช๐ซ Prove: โณ ๐จ๐ฉ๐ซ โ โณ ๐ช๐ซ๐ฉ ฬ ฬ ฬ ฬ โฅ ๐ฉ๐ซ ฬ ฬ ฬ ฬ ฬ ๐จ๐ซ Given ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ โฅ ๐ฉ๐ช ๐ฉ๐ซ Given โณ ๐จ๐ฉ๐ซ is a right triangle Definition of perpendicular line segments โณ ๐ช๐ซ๐ฉ is a right triangle Definition of perpendicular line segments ๐จ๐ฉ = ๐ช๐ซ Given ๐ฉ๐ซ = ๐ฉ๐ซ Reflexive property โณ ๐จ๐ฉ๐ซ โ โณ ๐ช๐ซ๐ฉ HL Exit Ticket (5 minutes) Lesson 25: Date: Congruence Criteria for TrianglesโAAS and HL 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 207 Lesson 25 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Name ___________________________________________________ Date____________________ Lesson 25: Congruence Criteria for TrianglesโAAS and HL Exit Ticket 1. Sketch an example of two triangles that meet the AAA criteria but are not congruent. 2. Sketch an example of two triangles that meet the SSA criteria that are not congruent. Lesson 25: Date: Congruence Criteria for TrianglesโAAS and HL 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 208 Lesson 25 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Exit Ticket Sample Solutions 1. Sketch an example of two triangles that meet the AAA criteria but are not congruent. Responses should look something like the example below. 2. Sketch an example of two triangles that meet the SSA criteria that are not congruent. Responses should look something like the example below. 45o 45o 35o 110o 35o 110o Problem Set Sample Solutions Use your knowledge of triangle congruence criteria to write proofs for each of the following problems. 1. Given: ฬ ฬ ฬ ฬ , ๐ซ๐ฌ ฬ ฬ ฬ ฬ โฅ ๐ฌ๐ญ ฬ ฬ ฬ ฬ โฅ ๐ฉ๐ช ฬ ฬ ฬ ฬ โฅ ๐ฌ๐ญ ฬ ฬ ฬ ฬ , ๐ฉ๐ช ฬ ฬ ฬ ฬ , ๐จ๐ญ = ๐ซ๐ช ๐จ๐ฉ Prove: โณ ๐จ๐ฉ๐ช โ โณ ๐ซ๐ฌ๐ญ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ โฅ ๐ฉ๐ช ๐จ๐ฉ Given ฬ ฬ ฬ ฬ โฅ ๐ฌ๐ญ ฬ ฬ ฬ ฬ ๐ซ๐ฌ Given ฬ ฬ ฬ ฬ โฅ ๐ฌ๐ญ ฬ ฬ ฬ ฬ ๐ฉ๐ช Given ๐จ๐ญ = ๐ซ๐ช Given ๐โ ๐ฉ = ๐โ ๐ฌ = ๐๐° Definition of perpendicular lines ๐โ ๐ช = ๐โ ๐ญ When two parallel lines are cut by a transversal, the alternate interior angles are equal in measure ๐ญ๐ช = ๐ญ๐ช Reflexive property ๐จ๐ญ + ๐ญ๐ช = ๐ญ๐ช + ๐ช๐ซ Addition property of equality ๐จ๐ช = ๐ซ๐ญ Segment addition โณ ๐จ๐ฉ๐ช โ โณ ๐ซ๐ฌ๐ญ AAS Lesson 25: Date: Congruence Criteria for TrianglesโAAS and HL 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 209 Lesson 25 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 2. 3. In the figure, ฬ ฬ ฬ ฬ ๐ท๐จ โฅ ฬ ฬ ฬ ฬ ๐จ๐น and ฬ ฬ ฬ ฬ ๐ท๐ฉ โฅ ฬ ฬ ฬ ฬ ๐น๐ฉ and ๐น is equidistant from โก๐ท๐จ and โก๐ท๐ฉ. Prove that ฬ ฬ ฬ ฬ ๐ท๐น bisects โ ๐จ๐ท๐ฉ. ฬ ฬ ฬ ฬ ๐ท๐จ โฅ ฬ ฬ ฬ ฬ ๐จ๐น Given ฬ ฬ ฬ ฬ โฅ ๐ฉ๐น ฬ ฬ ฬ ฬ ๐ท๐ฉ Given ๐น๐จ = ๐น๐ฉ Given ๐ฆโ ๐จ = ๐ฆโ ๐น = ๐๐° Definition of perpendicular lines โณ ๐ท๐จ๐น, โณ ๐ท๐ฉ๐น are right triangles Definition of right triangle ๐ท๐น = ๐ท๐น Reflexive property โณ ๐ท๐จ๐น โ โณ ๐ท๐ฉ๐น HL โ ๐จ๐ท๐น โ โ ๐น๐ท๐ฉ Corresponding angles of congruent triangles are congruent. ฬ ฬ ฬ ฬ bisects โ ๐จ๐ท๐ฉ ๐ท๐น Definition of an angle bisector ฬ ฬ ฬ ฬ โ ๐จ โ โ ๐ท, โ ๐ฉ โ โ ๐น, ๐พ is the midpoint of ๐จ๐ท ฬ ฬ ฬ ฬ ฬ โ ๐ฉ๐พ ฬ ฬ ฬ ฬ ฬ ๐น๐พ Given: Prove: 4. โ ๐จ โ โ ๐ท Given โ ๐ฉ โ โ ๐น Given ฬ ฬ ฬ ฬ ๐พ is the midpoint of ๐จ๐ท Given ๐จ๐พ = ๐ท๐พ Definition of midpoint โณ ๐จ๐พ๐ฉ โ โณ ๐ท๐พ๐น AAS ฬ ฬ ฬ ฬ ฬ ๐น๐พ โ ฬ ฬ ฬ ฬ ฬ ๐ฉ๐พ Corresponding sides of congruent triangles are congruent. Given: ๐ฉ๐น = ๐ช๐ผ, rectangle ๐น๐บ๐ป๐ผ Prove: โณ ๐จ๐น๐ผ is isosceles ๐ฉ๐น = ๐ช๐ผ Given Rectangle ๐น๐บ๐ป๐ผ Given ฬ ฬ ฬ ฬ โฅ ๐น๐ผ ฬ ฬ ฬ ฬ ๐ฉ๐ช Definition of a rectangle ๐ฆโ ๐น๐ฉ๐บ = ๐ฆโ ๐จ๐น๐ผ When two para. lines are cut by a trans., the corr. angles are equal in measure ๐ฆโ ๐ผ๐ช๐ป = ๐ฆโ ๐จ๐ผ๐น When two para. lines are cut by a trans., the corr. angles are equal in measure ๐ฆโ ๐น๐บ๐ป = ๐๐°, ๐ฆโ ๐ผ๐ป๐บ = ๐๐° Definition of a rectangle ๐ฆโ ๐น๐บ๐ฉ + ๐โ ๐น๐บ๐ป = ๐๐๐ Linear pairs form supplementary angles. ๐ฆโ ๐ผ๐ป๐ช + ๐โ ๐ผ๐ป๐บ = ๐๐๐ Linear pairs form supplementary angles. ๐ฆโ ๐น๐บ๐ฉ = ๐๐°, ๐โ ๐ผ๐ป๐ช = ๐๐° Subtraction property of equality โณ ๐ฉ๐น๐บ and โณ ๐ป๐ผ๐ช are right triangles Definition of right triangle ๐น๐บ = ๐ผ๐ป Definition of a rectangle โณ ๐ฉ๐น๐บ โ โณ ๐ป๐ผ๐ช HL ๐ฆโ ๐น๐ฉ๐บ = ๐ฆโ ๐ผ๐ช๐ป Corresponding angles of congruent triangles are equal in measure ๐ฆโ ๐จ๐น๐ผ = ๐ฆโ ๐จ๐ผ๐น Substitution property of equality โณ ๐จ๐น๐ผ is isosceles If two angles in a triangle are equal in measure, then it is isosceles. Lesson 25: Date: Congruence Criteria for TrianglesโAAS and HL 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 210