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Geometry Module 1, Topic D: Congruence

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Geometry Module 1, Topic D: Congruence


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Type: Lesson/Unit Plan


In Module 1, Topic D, students use the knowledge of rigid motions developed in Topic C to determine and prove triangle congruence. At this point, students have a well-developed definition of congruence supported by empirical investigation. They can now develop an understanding of traditional congruence criteria for triangles, such as SAS, ASA, and SSS, and devise formal methods of proof by direct use of transformations. As students prove congruence using the three criteria, they investigate why AAS also leads toward a viable proof of congruence and why they cannot use SSA to establish congruence. Examining and establishing these methods of proving congruency leads to analysis and application of specific properties of lines, angles, and polygons in Topic E.

Content Standard(s):
MA2015 (2016)
Grade: 9-12
8 ) Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions. [G-CO8]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.8- Given two congruent triangles and angle measures of one of the triangles, identify the angle measures of the other triangle.

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:
  • 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)
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.
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.
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.)

Tags: angles, ASA, congruence, rigid motions, SAS, sides, SSS, triangles
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There are six lessons in this topic.

This resource is free for teachers to access and use. All resources required for the lessons are available to print from the site.

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Author: Hannah Bradley