Two figures are congruent if some sequence of rigid motions carries one exactly onto the other.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
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 2: 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 3: 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 4: Watch an Example
One step at a time.
Watch Leo Prove Congruence
Two triangles look alike but sit in different places and face differently.
- Step 1
He slides the first until one corner lands on the matching corner of the second.
Step 5: Your Turn
Practice makes it stick.
The Matching Side
Problem 1 of 2
Two congruent triangles. One has a side of 7. What is the matching side on the other?
The Matching Angle
Problem 2 of 2
Two congruent shapes. One has an angle of 55 degrees. What is the matching angle?
degrees
Congruent or Not
1 of 4
Congruent shapes, one side is 12. The matching side?
2 of 4
Congruent shapes, one angle is 90. The matching angle?
3 of 4
Can a reflection be part of proving congruence?
4 of 4
Does a dilation preserve congruence?
Step 6: 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?
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
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.