Lesson 25: Congruence Criteria for Triangles—AAS
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Lesson 25 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Lesson 25: Congruence Criteria for Triangles—AAS 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 neighbor’s. Given: ๐ซ๐ฌ = ๐ซ๐ฎ, ๐ฌ๐ญ = ๐ฎ๐ญ ฬ ฬ ฬ ฬ is the angle bisector of ∠๐ฌ๐ซ๐ฎ Prove: ๐ซ๐ญ ๐ซ๐ฌ = ๐ซ๐ฎ Given ๐ฌ๐ญ = ๐ฎ๐ญ Given ๐ซ๐ญ = ๐ซ๐ญ Reflexive property โณ ๐ซ๐ฌ๐ญ ≅ โณ ๐ซ๐ฎ๐ญ SSS ∠๐ฌ๐ซ๐ญ ≅ ∠๐ฎ๐ซ๐ญ Corresponding angles of congruent triangles are congruent. ฬ ฬ ฬ ฬ ๐ซ๐ญ is the angle bisector of ∠๐ฌ๐ซ๐ฎ. Definition of an angle bisector Exploratory Challenge (23 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 ensures congruence. ANGLE-ANGLE-SIDE TRIANGLE CONGRUENCE CRITERIA (AAS): Given two triangles โณ ๐จ๐ฉ๐ช and โณ ๐จ′๐ฉ′๐ช′. If ๐จ๐ฉ = ๐จ′๐ฉ′ (Side), ๐∠๐ฉ = ๐∠๐ฉ′ (Angle), and ๐∠๐ช = ๐∠๐ช′ (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 angle of each triangle? Lesson 25: Congruence Criteria for Triangles—AAS and HL This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 215 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 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 to state the congruence and AAS. HYPOTENUSE-LEG TRIANGLE CONGRUENCE CRITERIA (HL): Given two right triangles โณ ๐จ๐ฉ๐ช and โณ ๐จ′๐ฉ′๐ช′ with right angles ๐ฉ and ๐ฉ′. If ๐จ๐ฉ = ๐จ′๐ฉ′ (Leg) and ๐จ๐ช = ๐จ′๐ช′ (Hypotenuse), then the triangles are congruent. PROOF: As with some of our other proofs, we do not start at the very beginning, but imagine that a congruence exists so that triangles have been brought together such that ๐จ = ๐จ′ and ๐ช = ๐ช′; 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 ๐ฉ๐ฉ′. โณ ๐จ๐ฉ๐ฉ′ is isosceles by definition, and we can conclude that base angles ๐∠๐จ๐ฉ๐ฉ′ = ๐∠๐จ๐ฉ′๐ฉ. Since ∠๐ช๐ฉ๐ฉ′ and ∠๐ช๐ฉ′ ๐ฉ are both the complements of equal angle measures (∠๐จ๐ฉ๐ฉ′ and ∠๐จ๐ฉ′๐ฉ), they too are equal in measure. Furthermore, since ๐∠๐ช๐ฉ๐ฉ′ = ๐∠๐ช๐ฉ′ ๐ฉ, the sides of โณ ๐ช๐ฉ๐ฉ′ opposite them are equal in measure: ๐ฉ๐ช = ๐ฉ′๐ช′. Then, by SSS, we can conclude โณ ๐จ๐ฉ๐ช ≅ โณ ๐จ′๐ฉ′๐ช′. Note that when using the Hypotenuse-Leg triangle congruence criteria as a reason in a proof, you need only to state the congruence and HL. Lesson 25: Congruence Criteria for Triangles—AAS and HL This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 216 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 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 length 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 isn’t AAA categorized as congruence criteria? Is there any situation in which AAA 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: Congruence Criteria for Triangles—AAS and HL This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 217 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1 GEOMETRY Examples (8 minutes) Examples 1. 2. Given: ฬ ฬ ฬ ฬ ⊥ ๐ช๐ซ ฬ ฬ ฬ ฬ , ๐จ๐ฉ ฬ ฬ ฬ ฬ ⊥ ๐จ๐ซ ฬ ฬ ฬ ฬ , ๐∠๐ = ๐∠๐ ๐ฉ๐ช Prove: โณ ๐ฉ๐ช๐ซ ≅ โณ ๐ฉ๐จ๐ซ ๐∠๐ = ๐∠๐ Given ฬ ฬ ฬ ฬ ๐จ๐ฉ ⊥ ฬ ฬ ฬ ฬ ๐จ๐ซ Given ฬ ฬ ฬ ฬ ๐ฉ๐ช ⊥ ฬ ฬ ฬ ฬ ๐ช๐ซ Given ๐ฉ๐ซ = ๐ฉ๐ซ Reflexive property ๐∠๐ + ๐∠๐ช๐ซ๐ฉ = ๐๐๐° Linear pairs form supplementary angles. ๐∠๐ + ๐∠๐จ๐ซ๐ฉ = ๐๐๐° Linear pairs form supplementary angles. ๐∠๐ช๐ซ๐ฉ = ๐∠๐จ๐ซ๐ฉ If two angles are equal in measure, then their supplements are equal in measure. ๐∠๐ฉ๐ช๐ซ = ๐∠๐ฉ๐จ๐ซ = ๐๐° Definition of perpendicular line segments โณ ๐ฉ๐ช๐ซ ≅ โณ ๐ฉ๐จ๐ซ AAS Given: ฬ ฬ ฬ ฬ ๐จ๐ซ ⊥ ฬ ฬ ฬ ฬ ฬ ๐ฉ๐ซ, ฬ ฬ ฬ ฬ ฬ ๐ฉ๐ซ ⊥ ฬ ฬ ฬ ฬ ๐ฉ๐ช, ๐จ๐ฉ = ๐ช๐ซ Prove: โณ ๐จ๐ฉ๐ซ ≅ โณ ๐ช๐ซ๐ฉ ฬ ฬ ฬ ฬ ⊥ ๐ฉ๐ซ ฬ ฬ ฬ ฬ ฬ ๐จ๐ซ Given ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ⊥ 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 = ๐ซ๐ฉ Reflexive property โณ ๐จ๐ฉ๐ซ ≅ โณ ๐ช๐ซ๐ฉ HL Closing (2 minutes) ๏ง ANGLE-ANGLE-SIDE TRIANGLE CONGRUENCE CRITERIA (AAS): Given two triangles โณ ๐ด๐ต๐ถ and โณ ๐ด′๐ต′๐ถ′. If ๐ด๐ต = ๐ด′๐ต′ (Side), ๐∠๐ต = ๐∠๐ต′ (Angle), and ๐∠๐ถ = ๐∠๐ถ′ (Angle), then the triangles are congruent. ๏ง HYPOTENUSE-LEG TRIANGLE CONGRUENCE CRITERIA (HL): Given two right triangles โณ ๐ด๐ต๐ถ and โณ ๐ด′๐ต′๐ถ′ with right angles ๐ต and ๐ต′. If ๐ด๐ต = ๐ด′๐ต′ (Leg) and ๐ด๐ถ = ๐ด′๐ถ′ (Hypotenuse), then the triangles are congruent. ๏ง The AAS and HL criteria implies the existence of a congruence that maps one triangle onto the other. ๏ง Triangles that meet either the Angle-Angle-Angle or the Side-Side-Angle criteria do not guarantee congruence. Exit Ticket (5 minutes) Lesson 25: Congruence Criteria for Triangles—AAS and HL This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 218 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 25 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Name Date Lesson 25: Congruence Criteria for Triangles—AAS 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: Congruence Criteria for Triangles—AAS and HL This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 219 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 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. 45o 45o 35o 2. 110o 35o 110o Sketch an example of two triangles that meet the SSA criteria that are not congruent. Responses should look something like the example below. 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 ๐จ๐ฉ ⊥ ฬ ฬ ฬ ฬ ๐ฉ๐ช, ฬ ฬ ฬ ฬ ๐ซ๐ฌ ⊥ ฬ ฬ ฬ ฬ ๐ฌ๐ญ, ฬ ฬ ฬ ฬ ๐ฉ๐ช โฅ ฬ ฬ ฬ ฬ ๐ฌ๐ญ, ๐จ๐ญ = ๐ซ๐ช Prove: โณ ๐จ๐ฉ๐ช ≅ โณ ๐ซ๐ฌ๐ญ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ⊥ ๐ฉ๐ช ๐จ๐ฉ Given ฬ ฬ ฬ ฬ ⊥ ๐ฌ๐ญ ฬ ฬ ฬ ฬ ๐ซ๐ฌ Given ฬ ฬ ฬ ฬ โฅ ๐ฌ๐ญ ฬ ฬ ฬ ฬ ๐ฉ๐ช Given ๐จ๐ญ = ๐ซ๐ช Given ๐∠๐ฉ = ๐∠๐ฌ = ๐๐° Definition of perpendicular lines ๐∠๐ช = ๐∠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 ≅ โณ ๐ซ๐ฌ๐ญ AAS Lesson 25: Congruence Criteria for Triangles—AAS and HL This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 220 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1 GEOMETRY 2. 3. In the figure, ฬ ฬ ฬ ฬ ๐ท๐จ ⊥ ฬ ฬ ฬ ฬ ๐จ๐น and ฬ ฬ ฬ ฬ ๐ท๐ฉ ⊥ ฬ ฬ ฬ ฬ ๐น๐ฉ and ๐น is equidistant from โก๐ท๐จ and โก๐ท๐ฉ. Prove that ฬ ฬ ฬ ฬ ๐ท๐น bisects ∠๐จ๐ท๐ฉ. ฬ ฬ ฬ ฬ ๐ท๐จ ⊥ ฬ ฬ ฬ ฬ ๐จ๐น Given ฬ ฬ ฬ ฬ ⊥ ๐ฉ๐น ฬ ฬ ฬ ฬ ๐ท๐ฉ Given ๐น๐จ = ๐น๐ฉ Given ๐∠๐จ = ๐∠๐น = ๐๐° Definition of perpendicular lines โณ ๐ท๐จ๐น, โณ ๐ท๐ฉ๐น are right triangles. Definition of right triangle ๐ท๐น = ๐ท๐น Reflexive property โณ ๐ท๐จ๐น ≅ โณ ๐ท๐ฉ๐น HL ∠๐จ๐ท๐น ≅ ∠๐น๐ท๐ฉ Corresponding angles of congruent triangles are congruent. ฬ ฬ ฬ ฬ bisects ∠๐จ๐ท๐ฉ. ๐ท๐น Definition of an angle bisector ฬ ฬ ฬ ฬ ∠๐จ ≅ ∠๐ท, ∠๐ฉ ≅ ∠๐น, ๐พ is the midpoint of ๐จ๐ท ฬ ฬ ฬ ฬ ฬ ≅ ๐ฉ๐พ ฬ ฬ ฬ ฬ ฬ ๐น๐พ Given: Prove: 4. ∠๐จ ≅ ∠๐ท Given ∠๐ฉ ≅ ∠๐น Given ๐พ is the midpoint of ฬ ฬ ฬ ฬ ๐จ๐ท. Given ๐จ๐พ = ๐ท๐พ Definition of midpoint โณ ๐จ๐พ๐ฉ ≅ โณ ๐ท๐พ๐น AAS ฬ ฬ ฬ ฬ ฬ ≅ ๐ฉ๐พ ฬ ฬ ฬ ฬ ฬ ๐น๐พ 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 ๐∠๐น๐ฉ๐บ = ๐∠๐จ๐น๐ผ When two para. lines are cut by a trans., the corr. angles are equal in measure. ๐∠๐ผ๐ช๐ป = ๐∠๐จ๐ผ๐น When two para. lines are cut by a trans., the corr. angles are equal in measure. ๐∠๐น๐บ๐ป = ๐๐°, ๐∠๐ผ๐ป๐บ = ๐๐° Definition of a rectangle ๐∠๐น๐บ๐ฉ + ๐∠๐น๐บ๐ป = ๐๐๐ Linear pairs form supplementary angles. ๐∠๐ผ๐ป๐ช + ๐∠๐ผ๐ป๐บ = ๐๐๐ Linear pairs form supplementary angles. ๐∠๐น๐บ๐ฉ = ๐๐°, ๐∠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 ≅ โณ ๐ป๐ผ๐ช HL ๐∠๐น๐ฉ๐บ = ๐∠๐ผ๐ช๐ป Corresponding angles of congruent triangles are equal in measure. ๐∠๐จ๐น๐ผ = ๐∠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: Congruence Criteria for Triangles—AAS and HL This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 221 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
