ALEX Lesson Plan

Triangle Congruence with Rigid Motions

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This lesson provided by:
 Author: Morgan Boyd Organization: Retirement
General Lesson Information
 Lesson Plan ID: 35593 Title: Triangle Congruence with Rigid Motions Overview/Annotation: This lesson will provide instruction on proving triangles to be congruent using rigid motions. Using the concept of transformations, the students will be able to manipulate the triangle on the coordinate plane. When using the coordinate plane to test congruence, the triangle or other object will slide, rotate, or flip to map onto the other object. Sometimes, the student will use a combination of the transformations. This lesson results from the ALEX Resource Gap Project.
Associated Standards and Objectives
Content Standard(s):
 Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis 25. Verify criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another. a. Verify that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. b. Verify that two triangles are congruent if (but not only if) the following groups of corresponding parts are congruent: angle-side-angle (ASA), side-angle-side (SAS), side-side-side (SSS), and angle-angle-side (AAS). Example: Given two triangles with two pairs of congruent corresponding sides and a pair of congruent included angles, show that there must be a sequence of rigid motions will map one onto the other. Unpacked Content Evidence Of Student Attainment:Students: Given a triangle and its image under a sequence of rigid motions (translations, reflections, and translations), verify that corresponding sides and corresponding angles are congruent. Given two triangles that have the same side lengths and angle measures, find a sequence of rigid motions that will map one onto the other. Use rigid motions and the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof to establish that the usual triangle congruence criteria make sense and can then be used to prove other theorems.Teacher Vocabulary:Corresponding sides and angles Rigid motions If and only if Triangle congruence Angle-Side-Angle (ASA) Side-Angle-Side (SAS) Side-Side->Side (SSS)Knowledge:Students know: Characteristics of translations, rotations, and reflections including the definition of congruence. Techniques for producing images under transformations. Geometric terminology which describes the series of steps necessary to produce a rotation, reflection, or translation. Basic properties of rigid motions (that they preserve distance and angle). Methods for presenting logical reasoning using assumed understandings to justify subsequent results.Skills:Students are able to: Use geometric descriptions of rigid motions to accurately perform these transformations on objects. Communicate the results of performing transformations on objects. Use logical reasoning to connect geometric ideas to justify other results. Perform rigid motions of geometric figures. Determine whether two plane figures are congruent by showing whether they coincide when superimposed by means of a sequence of rigid motions (translation, reflection, or rotation). Identify two triangles as congruent if the lengths of corresponding sides are equal (SSS criterion), if the lengths of two pairs of corresponding sides and the measures of the corresponding angles between them are equal (SAS criterion), or if two pairs of corresponding angles are congruent and the lengths of the corresponding sides between them are equal (ASA criterion). Apply the SSS, SAS, and ASA criteria to verify whether or not two triangles are congruent.Understanding:Students understand that: If a series of translations, rotations, and reflections can be described that transforms one object exactly to a second object, the objects are congruent. It is beneficial to have minimal sets of requirements to justify geometric results (e.g., use ASA, SAS, or SSS instead of all sides and all angles for congruence).Diverse Learning Needs: Essential Skills:Learning Objectives: GEO.25.1: Define congruent, corresponding, triangles, angles, and the concept of if and only if. GEO.25.2: Compare angles and sides of two triangles to determine congruency. GEO.25.3: Determine the lengths of sides and the measures of angles in triangles. GEO.25.4: Identify corresponding parts of triangles. Prior Knowledge Skills:Define congruent and sequence. Identify congruent figures. Recognize attributes of geometric shapes. Identify the length between vertices on a coordinate plane. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)

Local/National Standards:

Primary Learning Objective(s):

The student will be able to explain the techniques of transformations to prove triangles congruent.

The student will be able to prove triangles congruent by SSS, ASA, SAS or AAS.

The student will be able to explain the differences of each type of transformation.

The student will be able to combine transformations to prove triangles are congruent.

Preparation Information
 Total Duration: 31 to 60 Minutes Materials and Resources: PowerPoint Presentation that will explain step by step during the class: "Congruent Triangles" (see attached presentation)PencilsNotebook paperScissors (if necessary)Graph Paper (see attached file, make copies as needed)Exit Slip Transformations with Answers (see attached file make copies for all students)"SSS, SAS, ASA, and AAS congruences combined" and "All transformations combined" from Kuta Software (make copies for each student)https://www.khanacademy.org/math/geometry/hs-geo-congruence/hs-geo-triangle-congruence/v/finding-congruent-triangleshttp://www.watchknowlearn.org/Video.aspx?VideoID=18777 Technology Resources Needed: Desktop computer with PowerPoint softwareProjector and/or interactive whiteboard for displaying videos and PowerPointiPad, Chromebook, or MacBook for students to revisit websites used in classWebsites:Khan Academy "Congruent Triangles"-https://www.khanacademy.org/math/geometry/hs-geo-congruence/hs-geo-triangle-congruence/v/finding-congruent-trianglesWatch Know Learn- http://www.watchknowlearn.org/Video.aspx?VideoID=18777Kuta Software (worksheet generator)- https://www.kutasoftware.com/freeige.html. The triangle congruence and transformation worksheets are on this page. The worksheets will open in another tab. Background/Preparation: Teacher:The teacher will preview the PowerPoint presentation. The teacher needs to visit the website Kuta Software. Kuta Software has free worksheets that can be printed and copied. The teacher needs to preview the videos to determine when to stop and ask questions.Student:The student needs to be able to graph points on the coordinate plane. The student needs to be familiar with transformations and how triangles can be congruent. The three transformations are translation, reflection, and rotation. Three common ways to prove triangle congruence are as follows: Side-Side-Side, Side-Angle-Side, and Angle-Side-Angle.
Procedures/Activities: