Two figures are congruent if some sequence of rigid motions carries one exactly onto the other.
Step 1: Picture It
Start with something you already know.
Stacking two playing cards
- Two cards from the same deck lie on a table.
- One lies sideways and face up, and the other lies face down.
- Lift one, turn it or flip it, and set it on the other.
- It covers the other exactly, with no edge sticking out.
- A card from a giant deck has the same shape, but it never fits exactly.
Congruent means one figure can be moved to sit exactly on the other.
You may slide, turn or flip it, but you may never stretch or shrink it.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Triangle ABC is flipped across the dashed line. Its image is triangle DEF, and corner A lands on corner D. Corner A measures 35°. What does corner D measure, in degrees?
degrees
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Not just "looks the same"
The test is whether it can be moved onto the other, which is something you can actually do.
Several moves are allowed
A flip then a slide is fine. Any number of rigid motions, in any order.
What follows
Congruent figures have equal matching sides and equal matching angles, because rigid motions preserve both.
Step 3: Try It Yourself
Tap and try it out.
Every length and every angle is exactly as it was. The shape moved without changing, so the two figures are congruent.
Step 4: In Real Life
Puzzle pieces
Two puzzle pieces are congruent if one can be slid, turned or flipped to sit exactly on the other. Factories stamp thousands of congruent pieces from one mould.
Step 5: Watch an Example
One step at a time.
Watch Maya Land One Triangle on Another
Triangle ABC has corners A(1, 1), B(5, 1) and C(1, 4). Triangle DEF has corners D(7, −1), E(11, −1) and F(7, −4). Maya looks for a sequence of moves that carries the first exactly onto the second.
- Step 1
Maya’s first job is to say which corner should land on which. A is the square corner of the first triangle and D is the square corner of the second, so A should end up on D.
Watch Dev Read the Parts Off a Congruence
A flip and then a slide carry triangle JKL onto triangle MNP, so JKL ≅ MNP. Dev writes the first two steps; you supply the rest.
- Step 1
Dev starts from what the two moves promise. A flip keeps every length and every angle, and so does a slide, so MNP has the same measurements as JKL.
Watch Ana Turn Down a Pair That Looks Right
Two rectangles sit on the table. One is 6 cm by 4 cm. The other is 9 cm by 6 cm. Ana is asked whether they are congruent.
- Step 1
Ana’s first thought is that the two rectangles look alike, and she knows that is not the test. The test is whether a slide, a flip or a turn carries one exactly onto the other.
Step 6: Your Turn
Practice makes it stick.
The Garden Beds
Problem 1 of 17
A gardener builds two congruent triangular beds. She measures the first and finds sides of 9 feet, 6 feet and 5 feet. Edging goes all the way round both beds and costs $4 a foot. What does the edging cost, in dollars?
dollars
The Shelf Brackets
Problem 2 of 17
A carpenter cuts a bracket with sides of 12 inches, 5 inches and a long edge of 13 inches. For the other end of the shelf he cuts its mirror image. A metal strip goes along the long edge of each bracket only, and costs $2 an inch. What does the strip for both brackets cost, in dollars?
dollars
Matching Parts
Find the matching part first, then read off what is asked.
Problem 3 of 17
Triangle ABC ≅ triangle DEF. Side AB is 11 units. How long is side DE, in units?
units
Problem 4 of 17
Triangle ABC ≅ triangle DEF. The corner at B measures 52°. What does the corner at E measure, in degrees?
degrees
Problem 5 of 17
Triangle PQR ≅ triangle STU. Which side of STU matches side PR?
Problem 6 of 17
A figure is flipped, then slid, then turned. Is the last figure congruent to the one it started as?
Problem 7 of 17
The two triangles in the drawing are congruent, corner for corner as they are drawn. Side LM is 5 units. How long is side HJ, in units?
units
Whose Step?
One line in each piece of working is a slip. Find it, then put it right.
Problem 8 of 17
Priya was told that a triangle ABC lands on a triangle with corners D, E and F: A lands on E, B lands on F and C lands on D. She wrote the congruence statement. One line is a slip. Which?
Priya’s working
- 1A lands on E, B lands on F, C lands on D
- 2the matching pairs are A–E, B–F, C–D
- 3the second name is written in alphabetical order: D, E, F
- AnswerABC ≅ DEF
Problem 9 of 17
Fix Priya’s line 3. A lands on E, B lands on F and C lands on D. Complete the statement: ABC ≅ …
Problem 10 of 17
Leo was asked whether a rectangle 6 inches by 3 inches is congruent to a rectangle 12 inches by 6 inches. One line is a slip. Which?
Leo’s working
- 1rectangle A: 6 in by 3 in. rectangle B: 12 in by 6 in
- 2the two are the same shape, and that is what congruent means
- 3a sequence of rigid motions exists
- Answerthe two rectangles are congruent
Problem 11 of 17
Fix Leo’s line 2. Rectangle A is 6 inches by 3 inches. A rectangle congruent to A is turned on its side and put beside it. What is that rectangle’s perimeter, in inches?
inches
Problem 12 of 17
Ravi was asked for the length of EF, given that triangle ABC ≅ triangle DEF. One line is a slip. Which?
Ravi’s working
- 1ABC ≅ DEF, with AB = 9 cm, BC = 6 cm, CA = 7 cm
- 2E goes with B, and F goes with C
- 3so EF matches AB, which is 9 cm
- AnswerEF = 9 cm
Problem 13 of 17
Fix Ravi’s line 3. Triangle ABC ≅ triangle DEF, with AB = 9 cm, BC = 6 cm and CA = 7 cm. How long is side EF, in cm?
cm
Pick the Move
Decide which move carries the first figure onto the second before you work anything out.
Problem 14 of 17
The two triangles in the drawing are the same size. Which single move carries the first exactly onto the second?
Problem 15 of 17
These two triangles are the same size as well. Which single move carries the first exactly onto the second?
Same Size and Shape?
Select every option the question asks for. Nothing here needs working out.
Problem 16 of 17
Two figures are congruent. Select every statement that must be true.
Problem 17 of 17
Select every sequence that leaves a figure congruent to the one it started as.
Step 7: Quick Check
Show what you know.
Question 1 of 2
Congruent shapes, one side is 15. The matching side?
Question 2 of 2
How is congruence defined?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A triangle ABC is congruent to triangle BAC. That is, A goes with B, B goes with A, and C goes with itself. What must be true of the triangle?
Stretch 2 of 2
A rectangle is cut along the dashed line into two triangles. Are the two triangles congruent, and what carries one onto the other?
What You Learned
- Congruent means one can be carried onto the other by rigid motions.
- Several moves in sequence are allowed.
- Matching sides and angles are therefore equal.
For the grown-up
The misconception this lesson targets is that congruent means "looks the same". Read that way, a small rectangle and a large one of the same proportions get called congruent, and the child is left with no test to apply. Congruence here is something you do: find a slide, a flip, a turn, or a string of them, that lands one figure exactly on the other. Its cousin is the order of the letters. Writing ABC ≅ DEF claims A goes with D, B with E and C with F, and a child who writes the second name in alphabetical order has said something false without noticing. Asking "which corner lands on which?" out loud fixes both.
Congruent means reachable by rigid motions — Two figures are congruent if some sequence of translations, reflections and rotations carries one exactly onto the other. This is a stronger and more useful definition than "same shape and size", because it tells you how to check.
Corresponding parts are equal — If two figures are congruent, every matching pair of sides is equal and every matching pair of angles is equal. The word "corresponding" matters — you must identify which part maps to which before comparing anything.
The order of the letters carries information — Writing triangle ABC ≅ triangle DEF says A maps to D, B to E and C to F. The order is not decorative; it states the correspondence. A congruence written with the letters in the wrong order is a false statement.
Congruent is not the same as equal — Two distinct triangles can be congruent — they are different objects with identical measurements. "Equal" would mean they are the same triangle. The separate word exists because geometry cares about that distinction.
If one lands on the other — Two triangles. Can you slide, flip or turn one onto the other exactly? If yes, they are congruent. Congruent means same size and same shape, wherever they sit. You do not need to measure every side. One successful sequence of moves proves it.