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Congruence At a Glance Student Probe Is ∆ABC congruent to ∆DEF? How do you know? A F B C D E Answer: Yes, because the corresponding parts are congruent. This may be shown by measuring the sides and angles, or by using patty paper to trace and fit one triangle on the other. Lesson Description In this lesson students verify that two polygons are congruent using transformations (translations, reflections, or rotations). Students use patty paper to fit one geometric shape onto another to determine that the corresponding parts are congruent. Rationale What: Congruence in geometric shapes. Common Core State Standard: CC.9-12.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. Matched Arkansas Standard: AR.9-12.T.G.2.1 (T.2.G.1) Apply congruence (SSS ...) and similarity (AA ...) correspondences and properties of figures to find missing parts of geometric figures and provide logical justification Mathematical Practices: Reason abstractly and quantitatively. Model with mathematics. Attend to precision. Who: Students who do not understand congruence. Grade Level: Geometry Prerequisite Vocabulary: congruent, corresponding parts Prerequisite Skills: None Delivery Format: Individual, pairs or small groups Lesson Length: 15 min. Materials, Resources, Technology: patty paper, pencil, straight edge, geometry software (optional) Student Worksheets: Polygons (.pdf) To solve many mathematical problems, it is advantageous to recognize the relationships among abstract objects, such as numbers or functions. The study of congruence allows students to develop this manner of thinking by investigating the relationships among concrete objects. Additionally, the study of congruence can promote students’ development of visualization and spatial reasoning. Preparation Prepare copies of Polygons for each student. Provide patty paper and straight edges (rulers). Lesson The teacher says or does… Expect students to say or do… 1. Trace polygon ABCD on patty paper. Compare polygon ABCD to polygon A’B’C’D’. Trace polygon ABCD and place on top of polygon A’B’C’D’. 2. How do the polygons compare? Answers may vary, but listen for, “They are equal”, or “they are the same”, or “they are congruent”. 3. Let’s make a list of the AB and A’B’ parts of the two polygons BC and B’C’ that are the “same”. CD and C’D’ DA and D’A’ A and A ' If students do not, then the teacher says or does… Model for students. Are corresponding parts matched? Model for students, if necessary. Which parts “fit on top” of each other? Which parts “match”? Prompt students, if necessary. B and B ' C and C ' D and D ' 4. These “matching” parts are called corresponding parts. How can we tell that the They are the same measure. corresponding sides and angles are the same? 5. Polygons whose corresponding parts all have the same measures are called congruent. Mathematicians use this symbol for congruent: . 6. How do the areas of both Answers may vary, but listen polygons compare? for, “The areas are the same”. 7. Trace triangle ABC on patty paper and compare to triangle A’B’C’. Trace triangle ABC and place on top of triangle A’B’C’. Corresponding sides and angles are the same. Be certain that correct vertices and sides are matched. Congruent polygons are exact copies of each other. What is area? Is the same amount of space covered? Remind student to be precise with tracing and correctly label the original triangle. The teacher says or does… Expect students to say or do… 8. How do the triangles compare? How do you know? They are congruent. 9. Suppose you are told that DEF XYZ . What do you know about the triangles? (See Teacher Notes.) The corresponding sides and angles have the same measure. DE and XY EF andYZ DF and XZ D and X If students do not, then the teacher says or does… Are corresponding parts matched? Let’s list the corresponding parts. Let’s list the corresponding parts. What do we know about the corresponding parts of congruent polygons? E and Y F and Z have the same measure. 10. Repeat with additional polygons as needed. Teacher Notes 1. Congruence is a special case of similarity in that corresponding angles are congruent and corresponding sides are in a 1:1 ratio. 2. When two figures are congruent, one can be moved so that it fits exactly on the other. Those moves may be a translation, a reflection, or a rotation. 3. Congruent polygons have congruent corresponding parts—their matching sides and angles are congruent. 4. When naming congruent polygons, the corresponding vertices are always listed in the same order. Variations 1. Students may not recognize that when working with triangles it is not necessary to show that all six pairs of corresponding parts of the triangles are congruent in order to conclude that the triangles are congruent. This lesson might be extended by investigating which pairs of corresponding parts in triangles will determine congruence. 2. Use interactive geometry software such as Geometer’s Sketchpad, Cabri, or Geogebra for this lesson. Formative Assessment Is ∆ABC congruent to ∆DEF? How do you know? D E F A B C References Driscoll, M. (2007). Fostering Geometric Thinking. Portsmouth NH: Heinemann. Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide - Response to Intervention in Mathematics. Retrieved 2 25, 2011, from rti4sucess.