Congruent vs. Similarity — What's the Difference?
By Fiza Rafique & Maham Liaqat — Updated on May 15, 2024
Congruent figures are identical in shape and size, having equal corresponding angles and side lengths, whereas similar figures share the same shape but differ in size, with proportional corresponding sides.
Difference Between Congruent and Similarity
Table of Contents
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Key Differences
Congruent figures are exact copies of each other in both shape and size, meaning that one can be transformed into the other through rotations, reflections, or translations. Whereas similar figures only need to have corresponding angles equal and corresponding sides in proportion, not necessarily the same size.
When dealing with congruence, transformations do not alter the dimensions of the figures, thus all corresponding measurements remain equal. On the other hand, similarity allows for scaling, meaning the figures can be different sizes but must maintain the same shape.
In terms of geometric properties, congruent figures will always have the same area and perimeter as their counterparts. Whereas similar figures will have areas and perimeters that are scaled versions of each other, according to the square and linear ratios of their corresponding sides, respectively.
Congruence can be proven using several criteria in triangle geometry, such as Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Side-Angle (ASA). Whereas similarity in triangles can be shown using criteria like Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Angle (AA).
While congruent figures are essentially identical save for their orientation or position, similar figures can significantly differ in size, leading to various practical applications like in model scaling or architectural design where proportions are kept intact.
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Comparison Chart
Definition
Exact match in size and shape
Same shape, proportional sizes
Conditions
Equal corresponding sides and angles
Proportional corresponding sides, equal angles
Geometric Properties
Same area, perimeter
Areas and perimeters are scaled versions
Proving Criteria
SSS, SAS, ASA, AAS
SSS, SAS, AA
Transformations
Translation, rotation, reflection
Scaling, along with translation, rotation, reflection
Compare with Definitions
Congruent
Invariant under rotation, reflection, or translation.
A square and another square rotated 45 degrees are congruent.
Similarity
Same shape, different sizes.
Two circles of different diameters are similar.
Congruent
Applied in ensuring precision in manufacturing and design.
Machine parts are made congruent to ensure interchangeability.
Similarity
Scale variations do not affect shape properties.
Architectural models are built to be similar to the actual buildings.
Congruent
Always equal in area and perimeter.
Congruent shapes have identical measurements for their perimeters.
Similarity
Corresponding angles are equal, sides are proportional.
The sides of similar triangles are in the same ratio.
Congruent
Requires exact correspondence in geometric dimensions.
Congruent polygons fit perfectly over each other when overlaid.
Similarity
Helpful in creating maps and scale models.
A map is a smaller but similar representation of geographical areas.
Congruent
Corresponding; congruous.
Similarity
The quality or condition of being similar; resemblance.
Congruent
Coinciding exactly when superimposed
Congruent triangles.
Similarity
A corresponding aspect or feature; equivalence
A similarity of writing styles.
Congruent
Of or relating to two numbers that have the same remainder when divided by a third number. For example, 11 and 26 are congruent when the modulus is 5.
Similarity
Closeness of appearance to something else.
Congruent
Corresponding in character; congruous
Similarity
(philosophy) The relation of sharing properties.
Congruent
Harmonious.
Similarity
(maths) A transformation that preserves angles and the ratios of distances
Congruent
(mathematics) Having a difference divisible by a modulus.
Similarity
The property of two matrices being similar.
Congruent
(mathematics) Coinciding exactly when superimposed.
Similarity
The quality or state of being similar; likeness; resemblance; as, a similarity of features.
Hardly is there a similarity detected between two or three facts, than men hasten to extend it to all.
Congruent
(algebra) Satisfying a congruence relation.
Similarity
The quality of being similar
Congruent
Possessing congruity; suitable; agreeing; corresponding.
The congruent and harmonious fitting of parts in a sentence.
Similarity
A Getalt principle of organization holding that (other things being equal) parts of a stimulus field that are similar to each other tend to be perceived as belonging together as a unit
Congruent
Corresponding in character or kind
Similarity
Used in comparing growth rates in biology or economics.
The growth patterns of similar economic markets can be studied through scaling.
Congruent
Coinciding when superimposed
Congruent
Exactly matching in size and shape.
Two triangles are congruent if all three sides in one triangle are equal to the corresponding sides in the other.
Common Curiosities
What does it mean for two shapes to be congruent?
Two shapes are congruent if they are identical in shape and size and can be matched perfectly through movements like rotation or flipping.
Can you give an example of similar but not congruent figures?
Two rectangles with the same angle measurements but different side lengths are similar but not congruent.
What is necessary for two triangles to be similar?
Two triangles are similar if their corresponding angles are equal and their sides are in proportional lengths.
What are the practical applications of studying congruence and similarity?
In architecture and engineering, understanding congruence ensures that parts fit together correctly, while similarity is crucial for creating scale models and understanding different size systems without changing the shape.
How are similarity and congruence different?
Congruence involves exact matching in size and shape, while similarity involves shapes being the same only in shape but not necessarily in size.
How does scaling affect the properties of similar figures?
Scaling changes the size of the figures while maintaining their shape, altering properties like area and perimeter based on the scale factor.
Why is similarity important in teaching and learning geometry?
Understanding similarity helps students grasp concepts of proportionality and scale, which are applicable in various mathematical and real-world contexts.
Are there any specific fields where only congruence is applicable?
In fields requiring precise replication of items, such as manufacturing interchangeable parts or ensuring symmetry in artworks, congruence is necessary.
How do congruence and similarity relate to real-life measurements?
Congruence can be used in quality control processes to ensure products meet exact specifications, while similarity is used in fields like cartography where maps represent the real world at a reduced scale but maintain shape accuracy.
Can figures be both congruent and similar?
Yes, all congruent figures are inherently similar as they have identical shapes and proportions, but the reverse is not true.
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Written by
Fiza RafiqueFiza Rafique is a skilled content writer at AskDifference.com, where she meticulously refines and enhances written pieces. Drawing from her vast editorial expertise, Fiza ensures clarity, accuracy, and precision in every article. Passionate about language, she continually seeks to elevate the quality of content for readers worldwide.
Co-written by
Maham Liaqat
