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Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Lesson 24: Congruence Criteria for TrianglesβASA and SSS Classwork Opening Exercise Use the provided 30° 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 neighborβs. Does the use of a given angle measure guarantee that all the triangles constructed in class have corresponding sides of equal lengths? Lesson 24: Date: Congruence Criteria for TrianglesβASA and SSS 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. S.130 Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 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 π΄π΅πΆ and π΄β²π΅β²πΆβ². If mβ πΆπ΄π΅ = mβ πΆ β² π΄β² π΅β² (Angle), π΄π΅ = π΄β²π΅β² (Side), and mβ πΆπ΅π΄ = mβ πΆ β² π΅β² π΄β² (Angle), then the triangles are congruent. 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 β³ π΄π΅πΆβ²β²β² to β³ π΄π΅πΆ, we apply a reflection π across the line π΄π΅. A reflection will map π΄ to π΄ and π΅ to π΅, since they are on line π΄π΅. However, we will say that π(πΆβ²β²β²) = πΆ β . Though we know that π(πΆβ²β²β²) is now in the same halfplane of line π΄π΅ as πΆ, we cannot assume that πΆ β²β²β² maps to πΆ. So we have π(β³ π΄π΅πΆβ²β²β²) =β³ π΄π΅πΆ β . To prove the theorem, we need to verify that πΆ β is πΆ. By hypothesis, we know that β πΆπ΄π΅ β β πΆβ²β²β²π΄π΅ (recall that β πΆβ²β²β²π΄π΅ is the result of two rigid motions of β πΆ β² π΄β² π΅β² , so must have the same angle measure as β πΆ β² π΄β² π΅β² ). Similarly, β πΆπ΅π΄ β β πΆβ²β²β²π΅π΄. Since β πΆπ΄π΅ β π(β πΆ β²β²β² π΄π΅) β β πΆ β π΄π΅, and πΆ and πΆ β are in the same half-plane of line π΄π΅, we conclude that βββββ π΄πΆ and βββββββ π΄πΆ β must actually be the same ray. Because the β β βββββ , the point πΆ must be a point somewhere on π΄πΆ βββββ . Using the second equality points π΄ and πΆ define the same ray as π΄πΆ β²β²β² β β βββββ and βββββββ of angles, β πΆπ΅π΄ β π(β πΆ π΅π΄) β β πΆ π΅π΄, we can also conclude that π΅πΆ π΅πΆ must be the same ray. Therefore, the β β βββββ βββββ βββββ point πΆ must also be on π΅πΆ . Since πΆ is on both π΄πΆ and π΅πΆ , and the two rays only have one point in common, namely πΆ, we conclude that πΆ = πΆ β . We have now used a series of rigid motions to map two triangles onto one another that meet the ASA criteria. Lesson 24: Date: Congruence Criteria for TrianglesβASA and SSS 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. S.131 Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Side-Side-Side Triangle Congruence Criteria (SSS): Given two triangles π΄π΅πΆ and π΄βπ΅βπΆβ. If π΄π΅ = π΄β²π΅β² (Side), π΄πΆ = π΄β²πΆβ² (Side), and π΅πΆ = π΅β²πΆβ² (Side) then the triangles are congruent. Proof: Again, we do not need to 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 π΅ to π΅β², and label the angles created by the auxiliary line as π, π , π‘, and π’. Since π΄π΅ = π΄π΅β² and πΆπ΅ = πΆπ΅β², β³ π΄π΅π΅β² and β³ πΆπ΅π΅β² are both isosceles triangles respectively by definition. Therefore, π = π because they are base angles of an isosceles triangle π΄π΅π΅β². Similarly, mβ π‘ = mβ π’ because they are base angles of β³ πΆπ΅π΅β². Hence, β π΄π΅πΆ = mβ π + mβ π‘ = mβ π + mβ π’ = β π΄π΅β² πΆ. Since mβ π΄π΅πΆ = mβ π΄π΅β²πΆ, we say that β³ π΄π΅πΆ β β³ π΄π΅β²πΆ 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 βSSS.β Similarly, when using the Angle-Side-Angle congruence criteria in a proof, you need only state the congruence and βASA.β Now we have three triangle congruence criteria at our disposal: SAS, ASA, and SSS. We will use these criteria to determine whether or not pairs of triangles are congruent. Lesson 24: Date: Congruence Criteria for TrianglesβASA and SSS 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. S.132 Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 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: π is the midpoint of Μ Μ Μ Μ π»π , mβ π» = mβ π. 2. Given: Rectangle π½πΎπΏπ with diagonal πΎπ. 3. Given: π π = π π΅, π΄π = ππ . Lesson 24: Date: Congruence Criteria for TrianglesβASA and SSS 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. S.133 Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 4. Given: mβ π΄ = mβ π·, π΄πΈ = π·πΈ. 5. Given: π΄π΅ = π΄πΆ, π΅π· = π΄π΅, πΆπΈ = π΄πΆ. 1 4 Lesson 24: Date: 1 4 Congruence Criteria for TrianglesβASA and SSS 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. S.134 Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Problem Set Use your knowledge of triangle congruence criteria to write proofs for each of the following problems. 1. 2. 3. 4. Given: Circles with centers π΄ and π΅ intersect at πΆ and π·. Prove: β πΆπ΄π΅ β β π·π΄π΅. Given: Prove: β π½ β β π, π½π΄ = ππ΅, π½πΎ = πΎπΏ = πΏπ. Μ Μ Μ Μ πΎπ β Μ Μ Μ Μ πΏπ . Given: mβ π€ = mβ π₯ and mβ π¦ = mβ π§. Prove: (1) β³ π΄π΅πΈ β β³ π΄πΆπΈ Μ Μ Μ Μ β₯ π΅πΆ Μ Μ Μ Μ (2) π΄π΅ = π΄πΆ and π΄π· After completing the last exercise, Jeanne said, βWe also could have been given that β π€ β β π₯ and β π¦ β β π§. This would also have allowed us to prove thatβ³ π΄π΅πΈ β β³ π΄πΆπΈ.β Do you agree? Why or why not? Lesson 24: Date: Congruence Criteria for TrianglesβASA and SSS 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. S.135