Two figures are congruent when some sequence of rigid transformations carries one exactly onto the other.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why this definition
It is testable. "Same shape and size" is a description; "can be moved onto" is a procedure you can carry out.
What follows
Congruent figures have equal corresponding sides and equal corresponding angles, because rigid motions preserve both.
Order carries information
Writing triangle ABC congruent to triangle DEF says A matches D, B matches E and C matches F. The order is a claim.
The naming error
Listing the vertices in the wrong order makes a false statement, even when the two triangles genuinely are congruent.
Congruent is not similar
Similar figures have the same shape at any size. Congruent figures must be the same size too.
Congruence defined by movement
Two figures are congruent if some sequence of rigid motions carries one exactly onto the other. This definition is more useful than "same shape and size" because it tells you how to demonstrate it.
Corresponding parts are equal
Congruent figures have all matching sides and angles equal. Identifying which part corresponds to which is a prerequisite for any comparison, and rotated figures make it non-obvious.
The letter order is a claim
Triangle ABC ≅ triangle DEF states that A maps to D, B to E and C to F. Writing the letters in the wrong order makes the statement false even when the triangles are genuinely congruent.
Exhibiting a sequence is a proof
Naming a specific sequence of rigid motions that carries one figure onto the other establishes congruence completely. That is why this definition is preferred in the integrated sequence.
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: Watch an Example
One step at a time.
Watch Elena Read a Congruence Statement
Elena is told triangle ABC is congruent to triangle DEF, with AB = 5.
- Step 1
The naming order pairs A with D, B with E, and C with F.
Step 4: Your Turn
Practice makes it stick.
The Correspondence
Problem 1 of 2
Triangle ABC is congruent to triangle DEF, and AB = 5. What is DE?
The Angle
Problem 2 of 2
Same triangles, and angle B is 40°. What is angle E, in degrees?
degrees
Match the Parts
1 of 8
ABC congruent to DEF with BC = 9. What is EF?
2 of 8
Same triangles with angle A = 70°. What is angle D?
3 of 8
Same triangles with AC = 12. What is DF?
4 of 8
Do congruent figures have to be the same size? 1 yes, 0 no.
5 of 8
Angles of a triangle are 70 and 40. What is the third, in degrees?
6 of 8
Does the order of letters in a congruence statement matter? 1 yes, 0 no.
7 of 8
Which transformations can be used to show congruence?
8 of 8
ABC congruent to DEF with angle C = 55°. What is angle F?
Step 5: Quick Check
Show what you know.
Question 1 of 2
ABC congruent to DEF with AB = 7. What is DE?
Question 2 of 2
What does it mean for two figures to be congruent?
What You Learned
- Congruent means a sequence of rigid transformations maps one figure onto the other.
- Corresponding sides and angles of congruent figures are equal.
- The order of letters in a congruence statement states the correspondence.