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Association of Mathematics Teachers of NY State Kristin A. Camenga Houghton College November 7, 2013 Slides available at http://campus.houghton.edu/webs/employees/kcamenga/teachers.htm …you will learn how Common Core defines congruence. …you will use transformations to prove congruence in a variety of situations. …you should grow in confidence in the Common Core approach to congruence. …you will be ready to find more connections between transformations and what you already teach. G-CO.B: Understand congruence in terms of rigid motions G-CO.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G-CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. A rigid motion is a function of the plane that preserves angles and distances. All rigid motions are reflections, rotations, translations or some composition of the three. A rigid motion is also called an isometry, which means “same measure” With your neighbors, discuss the following problem: Show: A B 1. 𝐴𝐶 ≅ 𝐵𝐷 using rigid motions C D 2. ∠𝐴𝐶𝐷 ≅ ∠𝐵𝐷𝐶 What do students need to know to solve this problem? “Two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other.” Based on the definition of congruence, work with your neighbor to justify the following statement: Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Given: 𝐴𝐵≅ 𝐴′ 𝐵′ 𝐴𝐶 ≅ 𝐴′ 𝐶′ ∠𝐴 ≅ ∠𝐴′ We need to show a sequence of rigid motions that will map Δ𝐴𝐵𝐶 to Δ𝐴′ 𝐵′ 𝐶 ′ so all corresponding sides and angles coincide. Given: 𝐴𝐵≅ 𝐴′ 𝐵′, 𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐴 ≅ ∠𝐴′ Translate △ 𝐴𝐵𝐶 by vector 𝐴𝐴′ so that 𝐴 coincides with 𝐴’. Given: 𝐴𝐵≅ 𝐴′ 𝐵′, 𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐴 ≅ ∠𝐴′ Rotate △ 𝐴𝐵𝐶 by ∠𝐶𝐴𝐶′ around A so that ray 𝐴𝐶 coincides with ray 𝐴′ 𝐶′. Since 𝐴𝐵 ≅ 𝐴′ 𝐶′, 𝐶 coincides with 𝐶′. Given: 𝐴𝐵≅ 𝐴′ 𝐵′, 𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐴 ≅ ∠𝐴′ Reflect △ 𝐴𝐵𝐶 over 𝐴𝐵. Since ∠𝐴 ≅ ∠𝐴’ and the rays 𝐴𝐵 and 𝐴′ 𝐵′ coincide and are on the same side of the angle, ∠𝐴 coincides with∠𝐴’. Given: 𝐴𝐵≅ 𝐴′ 𝐵′, 𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐴 ≅ ∠𝐴′ Reflect △ 𝐴𝐵𝐶 over 𝐴𝐵. Since the angles coincide, the other rays 𝐴𝐶 and 𝐴′ 𝐶′ coincide. Since 𝐴𝐶 ≅ 𝐴′ 𝐶′, 𝐶 coincides with 𝐶′. Given: 𝐴𝐵≅ 𝐴′ 𝐵′, 𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐴 ≅ ∠𝐴′ Since all sides and angles coincide, △ 𝐴𝐵𝐶 ≅△ 𝐴′ 𝐵′ 𝐶 ′ . This proof guarantees that anytime we have SAS, there is a sequence of rigid motions that maps one of the triangles to the other, so they are congruent. Therefore, when using SAS, we do not need to use a sequence of rigid motions to show congruence. With your neighbors, find a set of rigid motions that will show that the following criteria are enough to prove triangle congruence. Make sure to explain how you know all the corresponding parts coincide! ASA SSS Given: ∠𝐴 ≅ ∠𝐴′, 𝐴𝐶 ≅ 𝐴′ 𝐶 ′ , ∠𝐶 ≅ ∠𝐶′ Translate & rotate as with SAS to align 𝐴 with 𝐴′ and 𝐶 with 𝐶′. If 𝐵 and 𝐵’ are on different sides of 𝐴𝐶, reflect △ 𝐴𝐵𝐶 over 𝐴𝐶. ◦ Since ∠𝐴 ≅ ∠𝐴’ and 𝐴𝐶 and 𝐴′ 𝐶′ coincide and are on the same side of the angle, ∠𝐴 coincides with ∠𝐴’. ◦ Since the angles coincide, the other rays 𝐴𝐵 and 𝐴′ 𝐵′ coincide. ◦ Similarly, since ∠𝐶 ≅ ∠𝐶’ and 𝐴𝐶and 𝐴′ 𝐶′ coincide, ∠𝐶 coincides with ∠𝐶’ and the other rays 𝐶𝐵 and 𝐶 ′ 𝐵′ coincide. ◦ Since the pairs of rays coincide, their intersections 𝐵 and 𝐵′ coincide. Since all sides and angles coincide, △ 𝐴𝐵𝐶 ≅△ 𝐴′𝐵′𝐶′. What is harder about SSS? No angles to align additional sides! What is harder about SSS? No angles to align additional sides! Draw BB ′ . What is harder about SSS? No angles to align additional sides! Then use SAS! For each of the following theorems, which transformations show the result? (Look for symmetries!) The base angles of an isosceles triangle are congruent. Vertical angles are congruent. A parallelogram has opposite sides and angles congruent. If a quadrilateral has diagonals that are perpendicular bisectors of each other, then it is a rhombus. If two lines are parallel, a transversal creates congruent alternate interior angles. Reflect over the angle bisector 𝐴𝐷. How does this compare to the standard proof using SAS? Rotation by 180◦ around the point of intersection Rotate 180◦ around the midpoint of one of the diagonals. 𝐴𝐵 ≅ 𝐵𝐶 𝐴𝐵 ≅ 𝐴𝐷 𝐵𝐶 ≅ 𝐶𝐷 So all four sides are congruent. Rotate about the midpoint of the transversal For each of the following quadrilaterals, describe the rotations and reflections that carry it onto itself: Parallelogram Rhombus Rectangle Square What connections do you notice? Parallelogram 180○ rotation Rhombus 180○ rotation, 2 lines of symmetry (diagonals) Rectangle 180○ rotation, 2 lines of symmetry (through midpoints of sides) Square 180○ rotation, 4 lines of symmetry Builds on students’ intuitive ideas so they can participate in proof from the beginning. Encourages visual and spatial thinking, helping students consider the same ideas in multiple ways. Serves as a guide for students to remember theorems and figure out problems. Reinforces properties of transformations and makes the geometry course more connected, both within itself and to algebra. Motivates changing perspective between pieceby-piece and global approaches. (MP.7) Ask students to look for symmetry regularly! When introducing transformations, apply them to common objects and ask what the symmetry implies about the object. Use transformations to organize information and remember relationships. Share another method of proof for a theorem already in your curriculum. The ideas of symmetry and transformation have application in algebra as well. This can help students connect algebra and geometry in a new way. … … … . … … … … . . … … … . . . … … … … Show mxn=nxm, Represent mxn as an array of dots with m rows and n columns. Rotate the array by 90 degrees and you have n rows and m columns, or nxm dots. Rotation preserves length & area, so these are the same number! Translations and reflections of graphs Odd & even functions Circles: x2 + y2 = r2 Unit circle trigonometry: sin(π/2-x) = cos(x) Questions? Kristin A. Camenga [email protected] Wallace, Edward C., and West, Stephen F., Roads to Geometry: section on transformational proof Henderson, David W., and Taimina, Daina, Experiencing Geometry NYS Common Core Mathematics Curriculum, Geometry: Module 1 These slides can be found at http://campus.houghton.edu/webs/employees/kcamenga/teachers.htm