Definition of Congruence

Examples, solutions, worksheets, videos, and lessons to help Grade 8 students know the definition of congruence and the properties for all three rigid motions (translations, rotations, and reflections).

New York State Common Core Math Grade 8, Module 2, Lesson 11

Worksheets for Grade 8

Student Outcomes

• Students know the definition of congruence and related notation, i.e., ≅ . Students know that to prove two figures are congruent there must be a sequence of rigid motions that maps one figure onto the other.
• Students know the basic properties of congruence are similar to the properties for all three rigid motions (translations, rotations, and reflections).

Congruence - the sequence of basic rigid motions that maps one figure onto another.
A sequence to show congruence can be any combination of translation, rotation and reflection.

In summary, if a figure S is congruent S’ then S’ is also congruent to S. In symbols S ≅ S’. It does not matter whether S comes first or S’ does.

Exercise 1
a. Describe the sequence of basic rigid motions that shows S₁ ≅ S₂
b. Describe the sequence of basic rigid motions that shows S₂ ≅ S₃
c. Describe the sequence of basic rigid motions that shows S₁ ≅ S₃

Basic properties of all three basic rigid motions

A basic rigid motion maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.

A basic rigid motion preserves lengths of segments.

A basic rigid motion preserves degrees of angles.

Exercise 2

Perform the sequence of a translation followed by a rotation of Figure XYZ, where T is a translation along a vector AB and R is a rotation of d degrees (you choose d) around a center O. Label the transformed figure X’Y’Z’. Will XYZ ≅ X’Y’Z'?

Lesson Summary:

Given that sequences enjoy the same basic properties of basic rigid motions, we can state three basic properties of congruences:

(C1) A congruence maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.

(C2) A congruence preserves lengths of segments.

(C3) A congruence preserves degrees of angles.

The notation used for congruence is ≅.

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