Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
In earlier grades, congruent meant 'same size and same shape', judged by eye. Grade 8 gives the word a precise meaning: one figure is congruent to another if some sequence of rotations, reflections and translations carries the first exactly onto the second. Because those moves preserve every length and angle (8.G.A.1), any figure produced this way matches the original part for part.
Students practice in both directions. Given two figures on a coordinate grid, they decide whether they are congruent and, if so, describe a sequence that maps one onto the other, for example 'reflect over the x-axis, then translate 3 units left'. Precision matters: a rotation needs a center, an angle and a direction, a reflection needs a line, and a translation needs a distance and direction. Often several different sequences work, and comparing them is a productive discussion. If the figures differ in size, no rigid sequence can match them, so they are not congruent, which sets up the contrast with similarity in 8.G.A.4.
Saying 'flip it' or 'turn it' is not enough. Students should name the line of reflection, or the center, angle and direction of rotation.
Two triangles can look identical but have one side slightly longer. Congruence needs a sequence that matches them exactly, or measurements that confirm it.
A figure and its mirror image are congruent even though one is reversed. Reflections are allowed in the sequence.
Reflecting then translating can give a different result from translating then reflecting. The order of a sequence matters and should be stated.
Triangle ABC has vertices A(1, 1), B(4, 1) and C(1, 3). Triangle A'B'C' has vertices A'(-1, -4), B'(-4, -4) and C'(-1, -6). Describe a sequence of rigid motions that maps ABC onto A'B'C'.
Answer: Rotate triangle ABC 180° about the origin, then translate it 3 units down. The triangles are congruent.
Give pairs of congruent figures on grids and challenge students to find the shortest sequence that works, then compare answers across groups. Tracing paper lets students test a guess physically. Including a pair that is not congruent, perhaps with one side stretched, reminds students that the definition can fail.
Assessment items often show two figures and ask which sequence maps one to the other, so students need fluency with coordinate rules for common moves. Ask them to verify a proposed sequence by tracking at least two vertices all the way through.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: C) One can be mapped onto the other by rotations, reflections and translations
Congruence means a sequence of rigid motions maps one figure exactly onto the other. Equal area or perimeter alone is not enough.
Answer: B) Yes, because both moves are rigid motions
Translations and reflections preserve lengths and angles, so any sequence of them produces a congruent figure.
Answer: -1
Moving 3 units left subtracts 3 from x: 2 - 3 = -1. The image is (-1, 1).
Answer: D) No, the side lengths are different
Rigid motions preserve lengths, so a triangle with sides 3, 4, 5 can never land exactly on one with sides 6, 8, 10. They are similar, not congruent.
Answer: -1
A 180° rotation about the origin changes the sign of both coordinates: (3, 1) maps to (-3, -1).
A full lesson with slides, activities and an exit ticket on congruence through rigid motions, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.G.A.2 with an answer key, ready in about a minute.
Make a worksheet →Turn congruence through rigid motions into a quiz students answer online that marks itself, with a class summary for you.
Build a test →No. Many different sequences can map one figure onto another. Any sequence that works is a valid description.
Yes. A reflection is a rigid motion, so a figure and its mirror image are congruent even though their orientation differs.