Lesson 24: Congruence Criteria for Triangles—ASA
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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1 GEOMETRY Lesson 24: Congruence Criteria for Triangles—ASA and SSS Student Outcomes ๏ง Students learn why any two triangles that satisfy the ASA or SSS congruence criteria must be congruent. Lesson Notes This is the third lesson in the congruency topic. So far, students have studied the SAS triangle congruence criteria and how to prove base angles of an isosceles triangle are congruent. Students examine two more triangle congruence criteria in this lesson: ASA and SSS. Each proof assumes the initial steps from the proof of SAS; ask students to refer to their notes on SAS to recall these steps before proceeding with the rest of the proof. Exercises require using all three triangle congruence criteria. Classwork Opening Exercise (7 minutes) Opening Exercise Use the provided ๐๐° angle as one base angle of an isosceles triangle. Use a compass and straight edge to construct an appropriate isosceles triangle around it. Compare your constructed isosceles triangle with a neighbor’s. Does using a given angle measure guarantee that all the triangles constructed in class have corresponding sides of equal lengths? No, side lengths may vary. Discussion (24 minutes) There are a variety of proofs of the ASA and SSS criteria. These follow from the SAS criteria already proved in Lesson 22. Discussion Today we are going to examine two more triangle congruence criteria, Angle-Side-Angle (ASA) and Side-Side-Side (SSS), to add to the SAS criteria we have already learned. We begin with the ASA criteria. ANGLE-SIDE-ANGLE TRIANGLE CONGRUENCE CRITERIA (ASA): Given two triangles โณ ๐จ๐ฉ๐ช and โณ ๐จ′๐ฉ′๐ช′. If ๐∠๐ช๐จ๐ฉ = ๐∠๐ช′ ๐จ′ ๐ฉ′ (Angle), ๐จ๐ฉ = ๐จ′๐ฉ′ (Side), and ๐∠๐ช๐ฉ๐จ = ๐∠๐ช′ ๐ฉ′ ๐จ′ (Angle), then the triangles are congruent. Lesson 24: Congruence Criteria for Triangles—ASA and SSS 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 208 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1 GEOMETRY PROOF: We do not begin at the very beginning of this proof. Revisit your notes on the SAS proof, and recall that there are three cases to consider when comparing two triangles. In the most general case, when comparing two distinct triangles, we translate one vertex to another (choose congruent corresponding angles). A rotation brings congruent, corresponding sides together. Since the ASA criteria allows for these steps, we begin here. In order to map โณ ๐จ๐ฉ๐ช′′′ to โณ ๐จ๐ฉ๐ช, we apply a reflection ๐ across the line ๐จ๐ฉ. A reflection maps ๐จ to ๐จ and ๐ฉ to ๐ฉ, since they are on line ๐จ๐ฉ. However, we say that ๐(๐ช′′′) = ๐ช∗. Though we know that ๐(๐ช′′′) is now in the same half-plane of line ๐จ๐ฉ as ๐ช, we cannot assume that ๐ช′′′ maps to ๐ช. So we have ๐(โณ ๐จ๐ฉ๐ช′′′ ) = โณ ๐จ๐ฉ๐ช∗. To prove the theorem, we need to verify that ๐ช∗ is ๐ช. By hypothesis, we know that ∠๐ช๐จ๐ฉ ≅ ∠๐ช′′′๐จ๐ฉ (recall that ∠๐ช′′′๐จ๐ฉ is the result of two rigid motions of ∠๐ช′ ๐จ′ ๐ฉ′ , so must have the same angle measure as ∠๐ช′ ๐จ′ ๐ฉ′ ). Similarly, ∠๐ช๐ฉ๐จ ≅ ∠๐ช′′′๐ฉ๐จ. Since ∠๐ช๐จ๐ฉ ≅ ๐(∠๐ช′′′ ๐จ๐ฉ) ≅ ∠๐ช∗ ๐จ๐ฉ, and ๐ช โโโโโ and โโโโโโโ and ๐ช∗ are in the same half-plane of line ๐จ๐ฉ, we conclude that ๐จ๐ช ๐จ๐ช∗ must actually be the same ray. Because the ∗ ∗ โโโโโ , the point ๐ช must be a point somewhere on ๐จ๐ช โโโโโ . Using the second equality of points ๐จ and ๐ช define the same ray as ๐จ๐ช angles, ∠๐ช๐ฉ๐จ ≅ ๐(∠๐ช′′′ ๐ฉ๐จ) ≅ ∠๐ช∗ ๐ฉ๐จ, we can also conclude that โโโโโโ ๐ฉ๐ช and โโโโโโโ ๐ฉ๐ช∗ must be the same ray. Therefore, the point ๐ช∗ must also be on โโโโโโ ๐ฉ๐ช. Since ๐ช∗ is on both โโโโโ ๐จ๐ช and โโโโโโ ๐ฉ๐ช, and the two rays only have one point in common, namely ๐ช, we conclude that ๐ช = ๐ช∗. We have now used a series of rigid motions to map two triangles onto one another that meet the ASA criteria. SIDE-SIDE-SIDE TRIANGLE CONGRUENCE CRITERIA (SSS): Given two triangles โณ ๐จ๐ฉ๐ช and โณ ๐จ′๐ฉ′๐ช′, if ๐จ๐ฉ = ๐จ′๐ฉ′ (Side), ๐จ๐ช = ๐จ′๐ช′ (Side), and ๐ฉ๐ช = ๐ฉ′๐ช′ (Side), then the triangles are congruent. PROOF: Again, we do not start at the beginning of this proof, but assume there is a congruence that brings a pair of corresponding sides together, namely the longest side of each triangle. Without any information about the angles of the triangles, we cannot perform a reflection as we have in the proofs for SAS and ASA. What can we do? First we add a construction: Draw an auxiliary line from ๐ฉ to ๐ฉ′, and label the angles created by the auxiliary line as ๐, ๐, ๐, and ๐. Lesson 24: Congruence Criteria for Triangles—ASA and SSS 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 209 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1 GEOMETRY Since ๐จ๐ฉ = ๐จ๐ฉ′ and ๐ช๐ฉ = ๐ช๐ฉ′, โณ ๐จ๐ฉ๐ฉ′ and โณ ๐ช๐ฉ๐ฉ′ are both isosceles triangles respectively by definition. Therefore, ๐ = ๐ because they are base angles of an isosceles triangle ๐จ๐ฉ๐ฉ′. Similarly, ๐∠๐ = ๐∠๐ because they are base angles of โณ ๐ช๐ฉ๐ฉ′. Hence, ∠๐จ๐ฉ๐ช = ๐∠๐ + ๐∠๐ = ๐∠๐ + ๐∠๐ = ∠๐จ๐ฉ′ ๐ช. Since ๐∠๐จ๐ฉ๐ช = ๐∠๐จ๐ฉ′๐ช, we say that โณ ๐จ๐ฉ๐ช ≅ โณ ๐จ๐ฉ′๐ช by SAS. We have now used a series of rigid motions and a construction to map two triangles that meet the SSS criteria onto one another. Note that when using the Side-Side-Side triangle congruence criteria as a reason in a proof, you need only state the congruence and SSS. Similarly, when using the Angle-Side-Angle congruence as a reason in a proof, you need only state the congruence and ASA. Now we have three triangle congruence criteria at our disposal: SAS, ASA, and SSS. We use these criteria to determine whether or not pairs of triangles are congruent. Exercises (6 minutes) These exercises involve applying the newly developed congruence criteria to a variety of diagrams. Exercises Based on the information provided, determine whether a congruence exists between triangles. If a congruence exists between triangles or if multiple congruencies exist, state the congruencies and the criteria used to determine them. 1. Given: ๐ด is the midpoint of ฬ ฬ ฬ ฬ ฬ ๐ฏ๐ท, ๐∠๐ฏ = ๐∠๐ท โณ ๐ฎ๐ฏ๐ด ≅ โณ ๐น๐ท๐ด, ASA 2. Given: ฬ ฬ ฬ ฬ ฬ Rectangle ๐ฑ๐ฒ๐ณ๐ด with diagonal ๐ฒ๐ด โณ ๐ฑ๐ฒ๐ด ≅ โณ ๐ณ๐ด๐ฒ, SSS/SAS/ASA 3. Given: ๐น๐ = ๐น๐ฉ, ๐จ๐น = ๐ฟ๐น โณ ๐จ๐น๐ ≅ โณ ๐ฟ๐น๐ฉ, SAS โณ ๐จ๐ฉ๐ ≅ โณ ๐ฟ๐๐ฉ, SAS Lesson 24: Congruence Criteria for Triangles—ASA and SSS 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 210 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1 GEOMETRY 4. ๐∠๐จ = ๐∠๐ซ, ๐จ๐ฌ = ๐ซ๐ฌ Given: โณ ๐จ๐ฌ๐ฉ ≅ โณ ๐ซ๐ฌ๐ช, ASA โณ ๐ซ๐ฉ๐ช ≅ โณ ๐จ๐ช๐ฉ, SAS/ASA 5. ๐ ๐ ๐ ๐ ๐จ๐ฉ = ๐จ๐ช, ๐ฉ๐ซ = ๐จ๐ฉ, ๐ช๐ฌ = ๐จ๐ช. Given: โณ ๐จ๐ฉ๐ฌ ≅ โณ ๐จ๐ช๐ซ, SAS Closing (1 minute) ๏ง ANGLE-SIDE-ANGLE TRIANGLE CONGRUENCE CRITERIA (ASA): Given two triangles โณ ๐ด๐ต๐ถ and โณ ๐ด′๐ต′๐ถ′. If ๐∠๐ถ๐ด๐ต = ๐∠๐ถ ′ ๐ด′ ๐ต′ (Angle), ๐ด๐ต = ๐ด′๐ต′ (Side), and ๐∠๐ถ๐ต๐ด = ๐∠๐ถ ′ ๐ต′ ๐ด′ (Angle), then the triangles are congruent. ๏ง SIDE-SIDE-SIDE TRIANGLE CONGRUENCE CRITERIA (SSS): Given two triangles โณ ๐ด๐ต๐ถ and โณ ๐ด′ ๐ต′ ๐ถ ′ . If ๐ด๐ต = ๐ด′ ๐ต′ (Side), ๐ด๐ถ = ๐ด′๐ถ′ (Side), and ๐ต๐ถ = ๐ต′๐ถ′ (Side), then the triangles are congruent. ๏ง The ASA and SSS criteria implies the existence of a congruence that maps one triangle onto the other. Exit Ticket (7 minutes) Lesson 24: Congruence Criteria for Triangles—ASA and SSS 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 211 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Name Date Lesson 24: Congruence Criteria for Triangles—ASA and SSS Exit Ticket Based on the information provided, determine whether a congruence exists between triangles. If a congruence exists between triangles or if multiple congruencies exist, state the congruencies and the criteria used to determine them. Given: ๐ต๐ท = ๐ถ๐ท, ๐ธ is the midpoint of ฬ ฬ ฬ ฬ ๐ต๐ถ Lesson 24: Congruence Criteria for Triangles—ASA and SSS 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 212 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Exit Ticket Sample Solutions Based on the information provided, determine whether a congruence exists between triangles. If a congruence exists between triangles or if multiple congruencies exist, state the congruencies and the criteria used to determine them. ฬ ฬ ฬ ฬ Given: ๐ฉ๐ซ = ๐ช๐ซ, ๐ฌ is the midpoint of ๐ฉ๐ช โณ ๐ฉ๐ซ๐ฌ ≅โณ ๐ช๐ซ๐ฌ, SSS โณ ๐จ๐ฉ๐ฌ ≅โณ ๐จ๐ช๐ฌ, SSS or SAS โณ ๐จ๐ฉ๐ซ ≅โณ ๐จ๐ช๐ซ, SSS or SAS Problem Set Sample Solutions Use your knowledge of triangle congruence criteria to write proofs for each of the following problems. 1. 2. Given: Circles with centers ๐จ and ๐ฉ intersect at ๐ช and ๐ซ Prove: ∠๐ช๐จ๐ฉ ≅ ∠๐ซ๐จ๐ฉ ๐ช๐จ = ๐ซ๐จ Radius of circle ๐ช๐ฉ = ๐ซ๐ฉ Radius of circle ๐จ๐ฉ = ๐จ๐ฉ Reflexive property โณ ๐ช๐จ๐ฉ ≅ โณ ๐ซ๐จ๐ฉ SSS ∠๐ช๐จ๐ฉ ≅ ∠๐ซ๐จ๐ฉ Corresponding angles of congruent triangles are congruent. ๐∠๐ฑ = ๐∠๐ด, ๐ฑ๐จ = ๐ด๐ฉ, ๐ฑ๐ฒ = ๐ฒ๐ณ = ๐ณ๐ด ฬ ฬ ฬ ฬ ฬ ≅ ๐ณ๐น ฬ ฬ ฬ ฬ ๐ฒ๐น Given: Prove: ๐∠๐ฑ = ๐∠๐ด Given ๐ฑ๐จ = ๐ด๐ฉ Given ๐ฑ๐ฒ = ๐ฒ๐ณ = ๐ณ๐ด Given ๐ฑ๐ณ = ๐ฑ๐ฒ + ๐ฒ๐ณ Partition property or segments add ๐ฒ๐ด = ๐ฒ๐ณ + ๐ณ๐ด Partition property or segments add ๐ฒ๐ณ = ๐ฒ๐ณ Reflexive property ๐ฑ๐ฒ + ๐ฒ๐ณ = ๐ฒ๐ณ + ๐ณ๐ด Addition property of equality ๐ฑ๐ณ = ๐ฒ๐ด Substitution property of equality โณ ๐จ๐ฑ๐ณ ≅ โณ ๐ฉ๐ด๐ฒ SAS ∠๐น๐ฒ๐ณ ≅ ∠๐น๐ณ๐ฒ Corresponding angles of congruent triangles are congruent. ฬ ฬ ฬ ฬ ฬ ≅ ๐ณ๐น ฬ ฬ ฬ ฬ ๐ฒ๐น If two angles of a triangle are congruent, the sides opposite those angles are congruent. Lesson 24: Congruence Criteria for Triangles—ASA and SSS 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 213 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 3. 4. Given: ๐∠๐ = ๐∠๐, and ๐∠๐ = ๐∠๐ Prove: (1) โณ ๐จ๐ฉ๐ฌ ≅ โณ ๐จ๐ช๐ฌ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ⊥ ๐ฉ๐ช (2) ๐จ๐ฉ = ๐จ๐ช and ๐จ๐ซ ๐∠๐ = ๐∠๐ Given ๐∠๐จ๐ฌ๐ฉ = ๐∠๐จ๐ฌ๐ช Supplements of equal angles are equal in measure. ๐จ๐ฌ = ๐จ๐ฌ Reflexive property ๐∠๐ = ๐∠๐ Given ∴โณ ๐จ๐ฉ๐ฌ ≅โณ ๐จ๐ช๐ฌ ASA ∴ ฬ ฬ ฬ ฬ ๐จ๐ฉ ≅ ฬ ฬ ฬ ฬ ๐จ๐ช Corresponding sides of congruent triangles are congruent. ๐จ๐ซ = ๐จ๐ซ Reflexive property โณ ๐ช๐จ๐ซ ≅ โณ ๐ฉ๐จ๐ซ SAS ๐∠๐จ๐ซ๐ช = ๐∠๐จ๐ซ๐ฉ Corresponding angles of congruent triangles are equal in measure. ๐∠๐จ๐ซ๐ช + ๐∠๐จ๐ซ๐ฉ = ๐๐๐° Linear pairs form supplementary angles. ๐(๐∠๐จ๐ซ๐ช) = ๐๐๐° Substitution property of equality ๐∠๐จ๐ซ๐ช = ๐๐° Division property of equality ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ⊥ ๐ฉ๐ช ๐จ๐ซ Definition of perpendicular lines After completing the last exercise, Jeanne said, “We also could have been given that ∠๐ ≅ ∠๐ and ∠๐ ≅ ∠๐. This would also have allowed us to prove that โณ ๐จ๐ฉ๐ฌ ≅ โณ ๐จ๐ช๐ฌ.” Do you agree? Why or why not? Yes; any time angles are equal in measure, we can also say that they are congruent. This is because rigid motions preserve angle measure; therefore, any time angles are equal in measure, there exists a sequence of rigid motions that carries one onto the other. Lesson 24: Congruence Criteria for Triangles—ASA and SSS 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 214 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
