8.G.A.2: Congruence Through Sequences of Rigid Motions
In plain English: 8.G.A.2 is the Common Core grade 8 math standard that defines congruent figures with rigid motions: one figure is congruent to another if a sequence of rotations, reflections and translations moves it exactly onto the other. Given two congruent figures, students describe such a sequence step by step. It is the Grade 8 Math meaning of congruence that high school geometry builds on.
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.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software. Also written as 8.G.2 · Official standard
In 8.G.A.1 students found that rigid motions (translations, reflections and rotations) never change lengths or angle measures. This lesson turns that fact into a definition. Two figures are congruent when a sequence of rigid motions, a list of moves done one after another, moves the first figure exactly onto the second. The symbol ≅ means "is congruent to".
Students work in both directions. First they apply a given sequence and see that the result matches the original in every length and angle. Then, given two congruent figures, they find and describe a sequence that moves one onto the other: which move, in which order, and with every detail (the distance and direction of a translation, the line of a reflection, the center, angle and direction of a rotation). They also learn to explain why two figures are not congruent: if a length or an angle has no match, no sequence can work.
Learning Objectives
By the end of this lesson, students will be able to:
Explain that two figures are congruent when a sequence of rotations, reflections and translations moves one exactly onto the other
Carry out a given sequence of rigid motions on a coordinate grid and name the matching vertices, sides and angles
Describe a sequence that maps one congruent figure onto another, with complete details for each move
Decide whether a reflection is needed by comparing the order of the vertices
Explain, using a length or an angle that has no match, why two figures are not congruent
Prior Knowledge Required
Students should already be comfortable with:
Rotations, reflections and translations, and the fact that they keep lengths, angle measures and parallel lines 8.G.A.1
Plotting points in all four quadrants of the coordinate plane 6.NS.C.8
Finding lengths of horizontal and vertical segments from coordinates 6.G.A.3
Give each pair two cut-out copies of the same paper triangle, one with no two sides equal. Tape one copy to the desk face down, so it is flipped compared with the other.
Warm-Up Prompt
"Move your loose triangle so it covers the taped triangle exactly. Write down each move you make, in order, using the words slide, flip and turn. Could you do it with slides and turns only?"
Pairs find that they must flip the loose triangle at some point; slides and turns alone never make it fit, because the taped copy is face down. Collect a few lists on the board and point out that different pairs used different lists that all worked. Tell students that such a list is called a sequence, and that today they will use sequences to define what it means for two figures to be congruent.
Direct Instruction20 minutes
Build these ideas on the board, with a transparency for each move. Students copy each one next to a sketch.
Sequence: a list of rigid motions done in order. Each move starts from the image made by the move before it. After the first move, points get one prime (A′); a second prime (A″) is optional, and the final figure can simply get new letters.
Congruent: one figure is congruent to another if some sequence of rotations, reflections and translations moves the first exactly onto the second. We write △ABC ≅ △DEF, listing the vertices in matching order: A goes to D, B to E and C to F.
Why congruent figures match: each rigid motion keeps every length and angle measure (8.G.A.1), so a whole sequence keeps them too. Matching sides, called corresponding sides, have equal lengths, and corresponding angles have equal measures.
Why some figures are not congruent: if one figure has a length or an angle that the other does not have, no sequence can work, because no rigid motion changes a length or an angle.
Finding a sequence: (1) match the vertices, using side lengths and angles; (2) compare the vertex order: if one figure goes around counterclockwise and the other clockwise, the sequence needs a reflection; (3) rotate so matching sides point the same way; (4) translate into place; (5) test the whole sequence with tracing paper.
Describing each move: a translation needs a distance and a direction, a reflection needs its line, and a rotation needs its center, its angle and clockwise or counterclockwise.
Reflection, then translation (Diagram 1)
Triangle ABC has A(-7, 2), B(-3, 2) and C(-6, 5). Triangle DEF has D(1, -2), E(5, -2) and F(2, -5). Show that they are congruent.
Equation: A to B to C goes counterclockwise, but D to E to F goes clockwise, so a reflection is needed. Reflect ABC across the x-axis: A′(-7, -2), B′(-3, -2), C′(-6, -5). Translate A′B′C′ 8 units right: (1, -2), (5, -2), (2, -5), which are D, E and F. So △ABC ≅ △DEF.
Rotation, then translation (Diagram 2)
Quadrilateral PQRS has P(1, 1), Q(5, 1), R(5, 3) and S(3, 4). Quadrilateral JKLM has J(-2, -6), K(-2, -2), L(-4, -2) and M(-5, -4). Describe a sequence that maps PQRS onto JKLM.
Equation: Both go around counterclockwise, so no reflection is needed. PQ is horizontal but JK is vertical, so start with a quarter turn: rotate 90° counterclockwise about the origin to get P′(-1, 1), Q′(-1, 5), R′(-3, 5), S′(-4, 3). Then translate 1 unit left and 7 units down onto J, K, L and M. So PQRS ≅ JKLM.
Two figures that are not congruent
Rectangle 1 has vertices (0, 0), (5, 0), (5, 3) and (0, 3). Rectangle 2 has vertices (0, 0), (0, 6), (-3, 6) and (-3, 0). Are they congruent?
Equation: No. Both have four right angles and a side of 3, but Rectangle 1 has sides of 5 and Rectangle 2 has sides of 6. No rigid motion turns a 5-unit side into a 6-unit side, so no sequence can map one rectangle onto the other.
More than one sequence works
Triangle GHI has G(2, 1), H(5, 1) and I(3, 4), and its partner has vertices (-2, -1), (-5, -1) and (-3, -4). Find two different sequences.
Equation: Sequence 1: rotate 180° about the origin, which sends (2, 1) to (-2, -1), (5, 1) to (-5, -1) and (3, 4) to (-3, -4). Sequence 2: reflect across the y-axis, giving (-2, 1), (-5, 1) and (-3, 4), then reflect across the x-axis. Both end on the same triangle, so either one shows the congruence.
Use Diagram 1 to show the in-between image A′B′C′ and to stress that the second move starts from it, not from ABC. Use the arc in Diagram 2 to show that during a rotation each point stays the same distance from the center.
Guided Practice15 minutes
Pairs work each problem with tracing paper. One partner proposes a sequence and the other tests it; they switch for the next problem.
Guided practice problems with answers
Problem
Answer
Segment 1 runs from (1, 2) to (1, 7). Segment 2 runs from (4, -3) to (9, -3). Are they congruent? If so, describe a sequence.
Yes, both are 5 units long. Rotate 90° clockwise about the origin to get (2, -1) and (7, -1), then translate 2 units right and 2 units down
Triangle 1 has vertices (1, 1), (3, 1) and (1, 4). Triangle 2 has vertices (-1, 1), (-3, 1) and (-1, 4). Which single move maps Triangle 1 onto Triangle 2?
A reflection across the y-axis. Each vertex lands the same distance from the y-axis on the other side
Triangle ABC has A(0, 0), B(4, 0) and C(0, 2). Triangle 2 has vertices (0, 0), (0, 4) and (-2, 0). Describe a sequence.
Rotate 90° counterclockwise about the origin: A stays at (0, 0), B goes to (0, 4) and C goes to (-2, 0). One move is enough
The triangle with vertices (0, 0), (2, 0) and (0, 2) and the triangle with vertices (0, 0), (4, 0) and (0, 4) have the same angles. Are they congruent?
No. Their legs (the two sides that form the right angle) are 2 and 4 units long, and no rigid motion changes a length. The larger one is a scaled copy, which is studied in 8.G.A.4
Listen for students who describe a move without its details ("rotate it"), and for students who do the second move from the original figure instead of from the first image.
Independent Practice10-15 minutes
Students work alone, then compare with a partner. Different correct sequences are expected.
Independent practice problems with answers
Problem
Answer
Triangle 1 has vertices (-5, -1), (-2, -1) and (-4, 2). Triangle 2 has vertices (1, 3), (4, 3) and (2, 6). Describe a sequence.
Translate 6 units right and 4 units up
Triangle 1 has vertices (2, 1), (6, 1) and (5, 3). Triangle 2 has vertices (-2, -1), (-6, -1) and (-5, -3). Describe a sequence.
Rotate 180° about the origin (or reflect across the y-axis, then across the x-axis)
One triangle has sides 5 cm, 5 cm and 6 cm. Another has sides 5 cm, 6 cm and 6 cm. Are they congruent?
No. The first has two 5 cm sides and the second has only one, so a length has no match
Give a sequence of two moves that takes the point (1, 3) to (-3, -1).
One answer: rotate 90° counterclockwise about the origin to (-3, 1), then reflect across the x-axis to (-3, -1)
A sequence maps R to Y, S to Z and T to X. Write the congruence statement.
△RST ≅ △YZX
Closure5 minutes
Exit ticket: (1) Describe a sequence that maps the segment from (0, 0) to (3, 0) onto the segment from (2, -1) to (2, -4). (Rotate 90° clockwise about the origin, then translate 2 units right and 1 unit down.) (2) Two segments measure 7 cm and 7.3 cm. Can a sequence of rigid motions map one onto the other? (No.) (3) In one sentence, explain why a figure and its image after any sequence of rigid motions are congruent.
Differentiation Strategies
For Struggling Students
Let students cut out the first figure and physically move it onto the second, naming each move aloud before writing it
Give a sentence frame for each move: "Translate ___ units ___ and ___ units ___", "Reflect across ___", "Rotate ___° ___ about ___"
Start with pairs that need one move, then two moves, before pairs where the order of the vertices has changed
For Advanced Students
Ask for the shortest possible sequence for each pair, and a reason no shorter one exists
Have students test whether a translation followed by a rotation always gives the same result as the rotation followed by the translation
Extension (beyond this standard): ask whether any sequence of rigid motions can be replaced by at most one reflection followed by one rotation and one translation, and test the idea on three pairs
Assessment Guidance
What to Look For
A complete answer lists the moves in order and gives every detail of each move, names the matching vertices, and checks the result with tracing paper or coordinates. For "not congruent" answers, look for a specific length or angle that has no match, not "they look different". Watch for students who say two figures with equal areas or equal angles must be congruent, who forget that the second move starts from the first image, and who write the congruence statement with the vertices out of order.
02
Classroom Activities
3 Activities
1
Secret Sequence
20 minPairs
Every challenge card shows the same start triangle KLM, with K(1, 1), L(4, 1) and M(1, 3), and a different end triangle. The card's back names the secret two-move sequence that made the end triangle. One partner reads the back; the other must find a sequence that works, which may be different, and test it with tracing paper.
The 3 Challenge Cards
Card 1: end triangle (-3, 0), (-3, 3), (-5, 0)
Card 2: end triangle (-2, -1), (1, -1), (-2, -3)
Card 3: end triangle (2, -3), (2, -6), (4, -3)
Secret Sequences (card backs)
Card 1: rotate 90° counterclockwise about the origin, then translate 2 units left and 1 unit down
Card 2: reflect across the x-axis, then translate 3 units left
Card 3: translate 2 units right and 1 unit up, then rotate 90° clockwise about the origin
Procedure
Match K, L and M to the end triangle's vertices using the side lengths 3, 2 and the slanted side
Check the vertex order to decide whether a reflection is needed
Write a full sequence, test it with tracing paper, then compare it with the card back
Discussion Questions
Only Card 2's end triangle goes around clockwise. Which move in its secret sequence caused that?
On Card 3, do the rotation first and the translation second. Do you still land on the end triangle?
Did anyone find a sequence different from a card back that still worked?
2
Congruent or Not?
15 minPairs
Pairs get 8 cards, each describing two figures. They sort the cards into Congruent and Not Congruent. For each congruent pair they describe a sequence; for each other pair they name the length or angle with no match.
The 8 Cards
Card A: two segments, each 6 cm long
Card B: a rectangle 4 cm by 7 cm, standing up, and a rectangle 7 cm by 4 cm, lying down
Card C: a square with 5 cm sides and a square with 5.5 cm sides
Card D: two triangles, each with angles 50°, 60° and 70°, whose shortest sides are 3 cm and 6 cm
Card E: the triangle with vertices (1, 1), (5, 1), (2, 3) and the triangle with vertices (-1, 1), (-5, 1), (-2, 3)
Card F: the triangle with vertices (1, 1), (5, 1), (2, 3) and the triangle with vertices (1, -1), (5, -1), (3, -3)
Card G: a rectangle with sides 3 cm and 5 cm, and a parallelogram with sides 3 cm and 5 cm and a 60° angle
Card H: two right triangles with legs 3 cm and 4 cm, one of them flipped over
Answer Key
Congruent: A (translate and rotate one onto the other), B (a quarter turn, then a translation), E (reflect across the y-axis), H (a reflection, then a rotation and a translation)
Not congruent: C (5 cm and 5.5 cm sides), D (3 cm and 6 cm shortest sides: one is a scaled copy of the other), F (the first triangle's slanted sides measure about 3.6 cm and 2.2 cm; the second's both measure about 2.8 cm), G (90° angles compared with a 60° angle)
Discussion Questions
Card F looks like a flip of the first triangle over the x-axis. How did you find out it is not?
Cards D and G each match in one way. What does each pair have in common, and why is that not enough?
3
Quilt Block Sequences
20 minGroups of 3
Each group gets a printed quilt block on a grid with four congruent triangles: M0 with vertices (1, 1), (4, 1), (1, 2); M1 with (-1, 1), (-1, 4), (-2, 1); M2 with (-1, -1), (-4, -1), (-1, -2); and M3 with (1, -1), (4, -1), (1, -2). The vertices of each triangle are listed in matching order. Groups describe a sequence from M0 to each other triangle, then one from M1 to M3.
Tasks
Describe a sequence that maps M0 onto M1, onto M2 and onto M3
Describe a sequence that maps M1 onto M3
Test every sequence with tracing paper
Answer Key
M0 to M1: rotate 90° counterclockwise about the origin
M0 to M2: rotate 180° about the origin
M0 to M3: reflect across the x-axis
M1 to M3: rotate 90° clockwise about the origin (back onto M0), then reflect across the x-axis. A single reflection across the diagonal line through (0, 0) and (1, 1) also works
Discussion Questions
M3 is the only triangle whose vertices go around clockwise. Why does every sequence from M0 to M3 include a reflection?
Quilters often turn one template to make a block. Which of the four triangles could a quilter make without flipping the template?
Design Variation
Groups design their own block with four copies of one motif, write the sequences on the back, and trade blocks with another group to check.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Reflection, Then a Translation
Triangle ABC with A(-7, 2), B(-3, 2) and C(-6, 5), drawn to scale. Step 1 reflects it across the x-axis to the dashed triangle A′B′C′. Step 2 translates A′B′C′ 8 units right onto triangle DEF, with D(1, -2), E(5, -2) and F(2, -5). The sequence shows that △ABC ≅ △DEF.
Diagram 2: A Rotation, Then a Translation
Quadrilateral PQRS with P(1, 1), Q(5, 1), R(5, 3) and S(3, 4), drawn to scale. Step 1 rotates it 90° counterclockwise about the origin to the dashed figure P′Q′R′S′; the arc shows S turning to S′ at a distance of 5 units from the center. Step 2 translates the figure 1 unit left and 7 units down onto JKLM.
04
Homework Assignment
~30 min
8.G.A.2 Homework: Showing Congruence with Rigid Motions
Directions: Use grid paper and tracing paper. For every sequence, describe each move completely and test the whole sequence. For every "not congruent" answer, name the length or angle that has no match.
Part 1: Describe a Sequence (Problems 1-3)
Triangle ABC has A(-5, 1), B(-2, 1) and C(-2, 3). Triangle DEF has D(2, -4), E(5, -4) and F(5, -2). (a) Describe a sequence that maps ABC onto DEF. (b) Write the congruence statement. (c) Check that AB = DE and BC = EF.
Triangle GHI has G(1, 2), H(1, 5) and I(3, 2). Triangle JKL has J(-2, -1), K(-5, -1) and L(-2, -3), with G matching J, H matching K and I matching L. (a) Compare the vertex order of the two triangles. What does it tell you? (b) Describe a sequence of two moves that maps GHI onto JKL.
Quadrilateral WXYZ has W(1, 1), X(5, 1), Y(4, 3) and Z(1, 3). Quadrilateral QRST has Q(6, 0), R(2, 0), S(3, -2) and T(6, -2). (a) Describe a sequence that maps WXYZ onto QRST, with W going to Q. (b) Which side of QRST matches side XY?
Part 2: Congruent or Not? (Problems 4-6)
Triangle 1 has sides 5 cm, 7 cm and 9 cm. Triangle 2 has sides 7 cm, 9 cm and 5 cm. Triangle 3 has sides 5 cm, 7 cm and 8 cm. (a) Which two triangles could be congruent? (b) Explain, using rigid motions, why the third cannot be congruent to either of them.
Omar says the triangle with vertices (0, 0), (5, 0), (0, 2) and the triangle with vertices (0, 0), (10, 0), (0, 4) are congruent because they have the same angles. (a) Is he right? (b) What kind of move would take one onto the other? (c) Why is that move not allowed in a sequence that shows congruence?
On a school map with 1 unit = 1 meter, the art room has corners (0, 0), (8, 0), (8, 6) and (0, 6). The music room has corners (10, -2), (16, -2), (16, 6) and (10, 6). (a) Show that the rooms are congruent by describing a sequence. (b) What are the length and width of each room?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Sequences
Moves listed in order with every detail, and the sequence works
Sequence works but a detail is missing
Missing or does not work
Matching Parts
Congruence statements and matching sides in the correct order
One matching error
No matching given
Not Congruent
Names the exact length or angle with no match
Says "different" without a specific measure
Missing or incorrect
Checking
Tests each sequence with tracing paper or coordinates
Tests some sequences
No testing shown
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order, using grid paper and tracing paper when they help. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Figure 2 is the image of Figure 1 after a reflection across the y-axis followed by a translation 4 units up. What can you conclude?
Answer: B
Any sequence of rotations, reflections and translations keeps every length and angle, so Figure 2 is congruent to Figure 1. Choice A treats a flip as a change of shape, but a reflection is a rigid motion. Choice D wrongly thinks a sequence must repeat one kind of move. Choice C adds a condition the definition does not have.
Question 2 of 20 · Multiple Choice
One triangle has sides 6 cm, 8 cm and 10 cm. Another has sides 6 cm, 8 cm and 11 cm. Can a sequence of rigid motions map one onto the other?
Answer: C
Rigid motions keep every length, so the 10 cm side would have to match a 10 cm side, and the second triangle has none. Choice A checks only two of the three sides. Choice B thinks a reflection can change a length. Choice D forgets that a translation can move a figure anywhere.
Question 3 of 20 · Multiple Choice
The triangle with vertices (2, 1), (5, 1) and (2, 3) is mapped onto the triangle with vertices (2, -1), (5, -1) and (2, -3), in that order. Which single move does this?
Answer: A
Each vertex keeps its x-coordinate and its y-coordinate changes sign, which is a reflection across the x-axis. Choice D works for (2, 1) but sends (2, 3) to (2, 1), not (2, -3): it checks only one vertex. Choice C would give (-2, -1), and choice B would give (-2, 1).
Question 4 of 20 · Multiple Choice
Point P(3, 2) is rotated 90° counterclockwise about the origin, then translated 5 units right. Where does it end up?
Answer: D
The rotation sends (3, 2) to (-2, 3), and the translation adds 5 to x: (3, 3). Choice A stops after the rotation. Choice B turns clockwise, to (2, -3), before the translation. Choice C does the moves in the wrong order: (8, 2) first, then the rotation.
Question 5 of 20 · Multiple Choice
△KLM ≅ △RST. Which side of △RST matches side KM?
Answer: B
The statement lists matching vertices in order: K with R, L with S and M with T. So KM matches RT. Choice A matches KL and choice C matches LM. Choice D is a side of the first triangle, not of △RST.
Question 6 of 20 · Multiple Choice
△KLM ≅ △RST, and angle L measures 72°. Which angle of △RST measures 72°?
Answer: C
L is the second vertex listed, so it matches S, the second vertex of △RST, and corresponding angles are equal: angle S = 72°. Choices A and B pair L with the wrong vertex. Choice D is an angle of △KLM, not of △RST.
Question 7 of 20 · Multiple Choice
The triangle with vertices (2, 1), (6, 1) and (3, 4) is congruent to the triangle with vertices (-3, -1), (1, -1) and (-2, -4), in that order. Which sequence maps the first triangle onto the second?
Answer: C
Going (2, 1) to (6, 1) to (3, 4) turns counterclockwise, but (-3, -1) to (1, -1) to (-2, -4) turns clockwise, so the sequence needs a reflection. Reflecting across the x-axis gives (2, -1), (6, -1) and (3, -4), and 5 units left gives (-3, -1), (1, -1) and (-2, -4). Choices A, B and D use only rotations and translations, which keep the vertex order, so they cannot work: A sends (3, 4) to (0, -4), B sends it to (-2, -1), and D sends (2, 1) to (-4, -2).
Question 8 of 20 · Multiple Choice
Which pair of figures is not congruent?
Answer: D
Both rectangles have an area of 12 cm², but their sides are 2 and 6 in one and 3 and 4 in the other, and rigid motions keep lengths. Choices A, B and C are congruent pairs: in each, a translation, followed by a rotation if needed, moves one figure exactly onto the other. Missing D is the error of thinking that equal area means congruent.
Question 9 of 20 · Multiple Choice
Which sequence maps the segment from (0, 0) to (4, 0) onto the segment from (1, 2) to (1, 6)?
Answer: A
The quarter turn counterclockwise sends (4, 0) to (0, 4), and the translation sends (0, 0) and (0, 4) to (1, 2) and (1, 6). Choice B turns the wrong way and ends at (1, 2) and (1, -2). Choice C does the moves in the wrong order and ends at (-2, 1) and (-2, 5). Choice D leaves the segment horizontal, from (1, 2) to (5, 2).
Question 10 of 20 · Multiple Choice
After a sequence of rigid motions, a triangle's image has a side of 7.5 cm and an angle of 40°. What must be true about the original triangle?
Answer: B
Every move keeps lengths and angle measures, so the original has the same 7.5 cm side and 40° angle. Choice A doubles the length. Choice C uses 180° - 40° and choice D uses 90° - 40°, as if a move changed the angle.
Question 11 of 20 · Multiple Choice
Point (-2, 5) is reflected across the x-axis and then translated 3 units left. Where does it end up?
Answer: A
The reflection gives (-2, -5), and 3 units left gives (-5, -5). Choice B moves right instead of left. Choice C reflects across the y-axis, to (2, 5), before moving left. Choice D skips the reflection.
Question 12 of 20 · Multiple Choice
A triangle is translated 3 units right, and then the image is translated 2 units down. Which single move gives the same result?
Answer: D
Every point moves 3 right and then 2 down, so each point ends 3 right and 2 down from where it started, which is one translation. Choice A subtracts the two distances, as if they were in the same direction. Choice C treats "down" as a flip. Choice B changes the direction the sides point, which neither translation does.
Question 13 of 20 · Multiple Choice
Which single move maps the triangle with vertices (1, 2), (4, 2) and (1, 4) onto the triangle with vertices (-2, 1), (-2, 4) and (-4, 1), in that order?
Answer: C
A quarter turn counterclockwise sends (1, 2) to (-2, 1), (4, 2) to (-2, 4) and (1, 4) to (-4, 1). Choice D works for the first vertex only: it sends (4, 2) to (1, 1). Choice A turns the wrong way, sending (1, 2) to (2, -1). Choice B sends (1, 2) to (-1, 2).
Question 14 of 20 · Multiple Choice
A triangle has an area of 12 cm². It is reflected and then rotated. What is the area of the final image?
Answer: A
The image is congruent to the original, so it has the same size and the same area, 12 cm². Choice B doubles the area because two moves were made. Choice C halves it, and choice D squares it; rigid motions do neither.
Question 15 of 20 · Short Answer
Triangle ABC has A(-6, 2), B(-3, 2) and C(-3, 4). Triangle DEF has D(6, -3), E(3, -3) and F(3, -1). Describe a sequence that maps ABC onto DEF, and write the congruence statement.
Reflect across the y-axis, giving (6, 2), (3, 2) and (3, 4), then translate 5 units down onto D, E and F. So △ABC ≅ △DEF. (Translating 5 units down first and then reflecting also works.)
Question 16 of 20 · Short Answer
One square has sides of 6 cm, and another has sides of 6.2 cm. Explain, using rigid motions, why they are not congruent.
Congruent means some sequence of rotations, reflections and translations moves one square exactly onto the other. Every one of those moves keeps lengths, so a 6 cm side stays 6 cm and can never cover a 6.2 cm side. No sequence works, so the squares are not congruent, even though all their angles match.
Question 17 of 20 · Short Answer
Triangle PQR has P(0, 1), Q(0, 5) and R(2, 1). Triangle STU has S(3, 0), T(7, 0) and U(3, -2), with P matching S, Q matching T and R matching U. Describe a sequence that maps PQR onto STU.
Rotate 90° clockwise about the origin: (0, 1) goes to (1, 0), (0, 5) to (5, 0) and (2, 1) to (1, -2). Then translate 2 units right onto S(3, 0), T(7, 0) and U(3, -2). Both triangles go around in the same direction, so no reflection is needed.
Question 18 of 20 · Short Answer
A sequence of rigid motions maps J to C, K to A and L to B. Write the congruence statement, and name the side of the second triangle that matches JL.
△JKL ≅ △CAB. J matches C and L matches B, so JL matches CB and the two sides have the same length.
Question 19 of 20 · Short Answer
Lucas says: "These two triangles are congruent. I can slide one onto the other, and then stretch it a little so it fits." Is he right? Explain.
No. A stretch changes lengths, so it is not a rotation, reflection or translation and cannot be part of a sequence that shows congruence. If the triangle needs a stretch to fit, some side lengths do not match, so the triangles are not congruent.
Question 20 of 20 · Short Answer
On a park map with 1 unit = 1 meter, a triangular flower bed has corners (1, 2), (5, 2) and (1, 5). A second bed has corners (-2, -1), (-2, -5) and (-5, -1), in matching order. Describe a sequence that maps the first bed onto the second.
One answer: rotate 90° clockwise about the origin, giving (2, -1), (2, -5) and (5, -1), then reflect across the y-axis onto (-2, -1), (-2, -5) and (-5, -1). A reflection is needed because the first bed's corners go around counterclockwise and the second bed's go clockwise. Both beds have legs of 4 m and 3 m.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.G.A.2 mean?
8.G.A.2 means that two figures are congruent when a sequence of rotations, reflections and translations moves one exactly onto the other. Students must understand this definition and, given two congruent figures, describe a sequence of moves that shows the congruence.
What is the grade 8 definition of congruent?
A figure is congruent to another if the second can be obtained from the first by a sequence of rigid motions. Earlier grades often say "same size and same shape"; grade 8 makes that precise, because rigid motions are exactly the moves that keep size and shape.
How do you describe a sequence of transformations completely?
List the moves in order and give every detail. A translation needs a distance and a direction, such as 4 units left and 2 units up. A reflection needs its line, such as the x-axis. A rotation needs its center, its angle and its direction, such as 90° clockwise about the origin.
Is there only one correct sequence in 8.G.A.2 problems?
No. Many different sequences can map one figure onto the same congruent figure, and any sequence that works is correct. Teachers should accept every tested sequence, and can ask students to compare two of them.
Does the order of the moves matter?
Often, yes. A rotation followed by a translation usually lands in a different place than the same translation followed by the same rotation. That is why students list the moves in order and do each move on the image from the move before.
What is the difference between congruent and similar figures?
Congruent figures have the same size and shape: rigid motions alone map one onto the other. Similar figures have the same shape but may differ in size, so a dilation (a scaling from a center) is allowed too. Similarity is the topic of 8.G.A.4.
Why does 8.G.A.2 define congruence with rigid motions?
Because "same size and same shape" is hard to check or to argue about. A sequence of moves is a concrete test: do the moves and see whether the figures match. It also explains why corresponding sides and angles of congruent figures are equal.
What mistakes do students make with congruence and sequences?
Many students leave out details such as the center of a rotation, start the second move from the original figure instead of from the first image, or write the congruence statement with the vertices out of order. Some think equal areas or equal angles are enough for congruence.
Is 8.G.A.2 only about triangles?
No. The standard covers any two-dimensional figure: segments, quadrilaterals, circles, letters and other shapes. Triangles are common in examples because they are easy to draw and to describe with coordinates.
How does 8.G.A.2 connect to high school geometry?
In high school, students use the same rigid-motion definition to decide whether figures are congruent (HSG.CO.B.6) and to show why side-and-angle tests for triangle congruence work. Grade 8 gives them the definition and the habit of describing moves precisely.
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Related Standards
5 standards
These standards connect to 8.G.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.G.A.1Prerequisite
Verify by experiment the properties of rotations, reflections and translations
