Congruence Through Rigid Motions

8.G.2Grade 8Geometry20 practice questions

Two figures are congruent when you can move one exactly onto the other using only slides, flips and turns. Same size, same shape - position doesn't matter.

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Similarity Through Dilations 8.G.4

What you'll be able to do

Describe a sequence of translations, reflections and rotations that maps one figure onto another.
Check a proposed sequence by moving every vertex.
Show two figures are not congruent by comparing side lengths.
Recognize a glide (a reflection followed by a slide).

The big idea

Translations, reflections and rotations are rigid motions: they move a figure without stretching or bending it. Every length and every angle stays the same.

So two figures are congruent if a sequence of rigid motions maps one onto the other. To find the sequence, look at orientation first: if the corners run the opposite way, you need a reflection. Then turn and slide.

To show two figures are not congruent, find one length that doesn't match. No rigid motion can change a length.

Words you need

Congruent (≅): Same size and shape: one maps exactly onto the other.
Rigid motion: A translation, reflection or rotation - it keeps every length and angle.
Glide reflection: A reflection followed by a translation along the mirror line, like footprints.
Corresponding parts: The sides and angles that land on each other.

Worked examples

Example 1

Triangle A has vertices (0, 0), (2, 0), (0, 3). Triangle B has vertices (0, 0), (0, 2), (−3, 0). Find a rigid motion that maps A onto B.

The side along the x-axis now lies along the y-axis, so try a quarter-turn.
Rotate 90° counterclockwise: (x, y) → (−y, x).
(0, 0) → (0, 0), (2, 0) → (0, 2), (0, 3) → (−3, 0). Every vertex matches.

Answer: Yes: (x, y) → (−y, x)

Example 2

Triangle A has vertices (1, 1), (4, 1), (1, 3). Triangle B has vertices (1, −1), (4, −1), (1, −3). Find a rigid motion that maps A onto B.

Each x-value stays the same and each y-value changes sign.
That's a reflection over the x-axis: (x, y) → (x, −y).

Answer: Yes: (x, y) → (x, −y)

Example 3

Is the triangle with vertices (0, 0), (3, 0), (0, 4) congruent to the one with vertices (0, 0), (6, 0), (0, 8)?

The first triangle's sides are 3, 4 and 5.
The second triangle's sides are 6, 8 and 10.
The lengths don't match, so no rigid motion maps one onto the other. Not congruent.

Answer: No, they are not congruent

Common mistakes

Checking only one vertex. A sequence works only if EVERY vertex lands on its partner. Check them all.

Calling figures congruent because they look alike. Same shape but a different size is similar, not congruent. Compare the lengths.

Forgetting a reflection when the corners run the other way. Slides and turns can't reverse orientation - only a flip can.

Practice: 20 questions

12 of 20 are word problems. For a word problem, sketch it first and mark what you know.

Tier 1 Foundations Use the rule directly

Find one rigid motion that maps triangle A with vertices (1, 1), (3, 1), (1, 4) onto triangle B with vertices (4, 1), (6, 1), (4, 4).
Find one rigid motion that maps triangle A with vertices (0, 1), (2, 1), (2, 4) onto triangle B with vertices (0, −1), (2, −1), (2, −4).
One sticker is a triangle with corners (0, 0), (4, 0), (0, 2). Another has corners (1, 3), (5, 3), (1, 5). Which single move maps the first onto the second? Word problem
A tile has corners (1, 0), (3, 0), (3, 2), (1, 2). A second tile has corners (−1, 0), (−3, 0), (−3, 2), (−1, 2). Which single move maps the first onto the second? Word problem
A sail is a triangle with corners (0, 0), (3, 0), (0, 4). A second sail has corners (0, 0), (4, 0), (0, 5). Are the sails congruent? Word problem

Tier 2 Intermediate One more step to think through

Find a rigid motion that maps triangle A with vertices (1, 2), (3, 2), (1, 5) onto triangle B with vertices (−2, 1), (−2, 3), (−5, 1).
Find a rigid motion that maps triangle A with vertices (2, 1), (5, 1), (2, 3) onto triangle B with vertices (−2, −1), (−5, −1), (−2, −3).
Find a sequence of two rigid motions that maps triangle A with vertices (1, 1), (3, 1), (1, 2) onto triangle B with vertices (−1, 4), (−3, 4), (−1, 5).
A garden bed is a triangle with corners (0, 0), (5, 0), (0, 12). Another bed has corners (1, 1), (6, 1), (1, 13). Show they are congruent with one move. Word problem
A logo with corners (2, 2), (4, 2), (2, 5) appears again with corners (−2, −2), (−4, −2), (−2, −5). Which single move maps one onto the other? Word problem
One kite has corners (0, 0), (2, 1), (4, 0), (2, −3). Another has corners (0, 0), (2, 1), (4, 0), (2, −4). Are the kites congruent? Word problem
A triangle has a side from (1, 1) to (4, 5). Ana makes a congruent copy. How long is the matching side of the copy? Word problem

Tier 3 Advanced Several steps - find the hidden structure

Find a rigid motion that maps triangle A with vertices (1, 1), (4, 1), (4, 3) onto triangle B with vertices (1, −1), (1, −4), (3, −4).
Find a sequence of two rigid motions that maps triangle A with vertices (0, 0), (2, 0), (0, 1) onto triangle B with vertices (3, 2), (3, 4), (2, 2).
A puzzle piece has corners (1, 2), (3, 2), (3, 3). It must fit a hole with corners (−2, −1), (−2, 1), (−3, 1). Find a sequence of two moves that fits it. Word problem
A left footprint has points (1, 1), (2, 1), (1, 3). The next footprint has points (4, −1), (5, −1), (4, −3). Find a sequence of two moves that maps one onto the next. Word problem
Two roof frames are triangles: one with corners (0, 0), (6, 0), (3, 4), the other with corners (0, 0), (8, 0), (4, 3). Are they congruent? Word problem

Tier 4 Challenge Stretch problems, some above grade level

Find a sequence of three rigid motions that maps triangle A with vertices (1, 0), (3, 0), (1, 1) onto triangle B with vertices (0, −3), (0, −5), (−1, −3).
Mia says that if two triangles have the same three angles, they must be congruent. Is she right? Word problem
A triangular flag has corners (2, 1), (4, 1), (2, 2). After a 90° counterclockwise rotation about the point (1, 1), where are its corners? Word problem Above grade level

Answers

Tier 1 Foundations

Yes: (x, y) → (x + 3, y). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
Yes: (x, y) → (x, −y). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
Yes: (x, y) → (x + 1, y + 3). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
Yes: (x, y) → (−x, y). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
No, they are not congruent. Rigid motions keep every length, and these shapes have different side lengths.

Tier 2 Intermediate

Yes: (x, y) → (−y, x). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
Yes: (x, y) → (−x, −y). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
Yes: (x, y) → (−x, y), then (x, y) → (x, y + 3). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
Yes: (x, y) → (x + 1, y + 1). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
Yes: (x, y) → (−x, −y). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
No, they are not congruent. Rigid motions keep every length, and these shapes have different side lengths.
5. Across 3, up or down 4: d² = 3² + 4² = 25, so d = 5.

Tier 3 Advanced

Yes: (x, y) → (y, −x). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
Yes: (x, y) → (−y, x), then (x, y) → (x + 3, y + 2). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
Yes: (x, y) → (−y, x), then (x, y) → (x, y − 2). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
Yes: (x, y) → (x, −y), then (x, y) → (x + 3, y). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
No, they are not congruent. Rigid motions keep every length, and these shapes have different side lengths.

Tier 4 Challenge

Yes: (x, y) → (−x, y), then (x, y) → (−y, x), then (x, y) → (x, y − 2). These steps are all rigid motions and they land every vertex on its partner, so the figures are congruent.
No. Equal angles only fix the shape. A larger copy has the same three angles but longer sides, so it isn't congruent.
(1, 2), (1, 4), (0, 2). Apply rotate 90° counterclockwise about (1, 1) to each point.

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