If we trace the word congruent back to its Latin roots, we find that it originally meant “to agree.”
That’s a good clue!
In everyday language, when things agree, they match up or line up just right. But what does that mean in geometry?
How can two shapes agree with each other? What exactly do they have in common? And how can we tell if two figures are congruent just by looking at them?
That’s what this guide is here to help you understand. Whether you're learning about congruent figures for the first time, reviewing before a test, or simply getting ahead in math, you're in the right place.
With easy-to-follow explanations, step-by-step examples, a fun quiz, and answers to questions students often ask, we’ll explore how congruent shapes work and why they matter.
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In geometry, congruent means that two figures are exactly the same in size and shape. If you were to pick one up and flip it, rotate it, or slide it over the other, they would match up perfectly, like two puzzle pieces cut from the same mold.
To be congruent, the figures must have matching side lengths and identical angles. It doesn’t matter if they’re in different positions. What matters is that every part lines up: side to side, angle to angle.
In geometry, we use the symbol ≅ to show that two figures are congruent.
This symbol has two parts:
The top is a tilde (~), which signals that the figures have the same shape.
The bottom is an equals sign (=), which shows they are the same size.
Placed together, ≅ means the figures are identical in both shape and size.
So for example, if we wanted to express that two triangles ABC and DEF are congruent, we would write:
△ABC ≅ △DEF
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Many students get confused by the words congruent and similar, and that’s understandable. These terms are closely related, but they don’t mean the same thing.
To be congruent, two figures must be exactly the same in both shape and size. Each side must match in length, and each angle must match in measure.
Similar figures, on the other hand, only need to have the same shape. Their sizes can be different. In similar figures:
All corresponding angles are equal.
The sides are in the same ratio, but not necessarily the same length.
Think of it like this:
All congruent figures are also similar, because they share the same shape and angle measures. But not all similar figures are congruent, because their sizes might be different.
So while congruent and similar figures can look alike, only congruent ones are true geometric twins.
In the diagram, rectangles ABCD and EFGH are congruent. They have the same shape and size—each side and angle matches exactly.
Rectangles IJKL and MNOP are similar. They have the same shape and angles, but their sides are not the same length.
In geometry, congruence is a concept we use when describing:
Line segments: Congruent segments are the same length.
Angles: Congruent angles have equal measure.
Polygons: Figures like triangles, rectangles, and other multi-sided shapes are congruent when all their sides and angles match.
Circles: Congruent circles have equal radii and are the same size overall.
These figures can match exactly when moved using geometric transformations, such as translations (slides), reflections (flips), and rotations (turns), without any change in size or shape.
Let’s take a look at how congruence shows up in these shapes.
Two line segments are congruent when they are the same length.
It doesn’t matter where they are on the page or how they’re positioned. If their lengths match exactly, they are congruent.
Take a look at the diagram below. Let’s say segment AB is 5 inches long, and segment CD is also 5 inches. Since they measure the same, we say:
AB ≅ CD
Two angles are congruent when they have the same degree measure.
It doesn’t matter how long their sides are or how the angles are rotated or placed on the page, if their angle measures are equal, they are congruent.
Let’s look at an example.
If we have angles ∠ABC and ∠DEF, and each measures 40°, no matter how they are positioned, we can write:
∠ABC ≅ ∠DEF
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Polygons—such as triangles, rectangles, parallelograms, pentagons, and hexagons—are considered congruent when they have the same number of sides, and all of their corresponding sides and angles are exactly equal.
Let’s say we have two triangles, △ABC and △DEF.
Since AB = DE, AC = DF, and BC = EF, the sides in both triangles are equal.
Also, the angles match:
∠A = ∠D = 90°,
∠B = ∠E = 45°,
∠C = ∠F = 45°.
So we can conclude:
△ABC ≅ △DEF
Circles are congruent when they have the same radius, which is the distance from the center to any point on the circle.
Why do we use the radius?
Because circles don’t have sides or angles like polygons do. The only way to compare their size is by measuring the radius. If the radii are equal, the circles are the same size, and since all circles have the same shape, that makes them congruent.
Take a look at the diagram.
We see two circles, ⊙A and ⊙B, each with a radius of 1.5 inches.
Since their radii are equal, the circles are the same size. That means:
⊙A ≅ ⊙B
Now, let’s think beyond the classroom. If we know that congruent shapes can be placed perfectly over each other, what would be some examples from real life?
The answers are plenty.
We see congruent figures in:
Floor tiles that fit together with no gaps or overlaps
Bicycle wheels that must be the same size to spin smoothly
Playing cards made to match exactly in shape and size
Building blocks that stack neatly because they're uniform
Notebook pages that line up edge to edge without sticking out
So next time you stack blocks, shuffle cards, or flip through your notebook, remember, you’re basically doing geometry.
Playing cards are congruent because they have the same size and shape. When one is placed over another, they line up perfectly.
Ready to put what you’ve learned to the test? Try our flash quiz below.
When you’re done, check your results at the bottom of this guide.
1. What does it mean for two figures to be congruent?
a) They have the same shape but not necessarily the same size
b) They have equal area
c) They are identical in both shape and size
d) They are mirror images
2. Circle A has a radius of 3 cm. Circle B also has a radius of 3 cm. What can we say about the two circles?
a) They are similar but not congruent
b) They are congruent
c) They have different sizes but the same shape
d) We can’t tell without knowing the diameter
3. A triangle is rotated 90 degrees but not resized. Is it still congruent to its original position?
a) Yes
b) No
4. Which of the following statements is correct?
a) All similar figures are also congruent.
b) Congruent figures are always similar.
c) Similar figures always have the same size.
d) Congruent figures must face the same direction.
5. Two angles each measure 40°. What can we say about them?
a) They are congruent.
b) They are supplementary.
c) They are vertical angles.
d) They must be part of the same triangle.
6. Triangles △PQR and △XYZ are congruent. If ∠P = 48°, what is the measure of the angle in △XYZ that corresponds to ∠P?
a) 32°
b) 48°
c) 84°
d) It depends on the side lengths
When students first encounter congruent as a term in their math classes, it can bring up quite a few questions.
We’ve compiled a list of the ones we often hear at Mathnasium of Plano Legacy West, along with clear answers to help make things click.
Students are typically introduced to the idea of congruence in upper elementary or early middle school, often around Grade 4 to Grade 6. They start with simple shapes and gradually move toward more complex figures and transformations in later grades.
Not always at first glance. Two figures can be rotated, flipped, or shifted and still be congruent. What matters is that their shape and size are the same, even if their positions differ.
Yes. Two shapes can cover the same amount of space but have different side lengths or angles. For example, a long rectangle and a square could have the same area but are not congruent.
Yes, as long as the shape and size are the same. Flipping a shape creates a reflection, which is a type of geometric transformation that preserves congruence.
In geometry, equal usually refers to specific values, like a side being 5 cm long. Congruent is used for entire figures, meaning all their corresponding parts (sides and angles) are equal.
Absolutely. Orientation doesn’t affect congruence. A triangle pointing left is congruent to one pointing right if all their sides and angles still match.
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Our specially trained tutors provide personalized in-center instruction and online support, guiding students to learn and master any math class and topic, including congruent figures, typically introduced in middle school math.
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Whether your student is looking to catch up, keep up, or get ahead in their math class, schedule an assessment and enroll at Mathnasium of Plano Legacy West today!
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If you’ve given our quiz a go, it’s time to check your answers. How many did you get right?
Question 1: c) They are identical in both shape and size.
Question 2: b) They are congruent.
Question 3: a) Yes
Question 4: b) Congruent figures are always similar
Question 5: a) They are congruent
Question 6: b) 48°
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