What is Congruence?

Congruence refers to two figures or objects that are exact replicas of one another. The congruent figures overlap point-to-point when they are stacked one on top of the other. Even though the figures may be in different planes, they will still be congruent if their sides, covered areas, and volume are all the same.1

The Significance of Congruence

Building our understanding of structures requires learning about congruence in figures. Due to the development of spatial intelligence, you may occasionally find two structures, objects, or products that are completely identical to one another. Congruent figures is a concept that is better understood with the same intelligence.

The concept of congruence is helpful in the fields of architecture, landscape design, product modeling, and business-to-business manufacturing. In addition to having a practical benefit, congruence strengthens learners’ foundations and helps them form a more fluid understanding of the concepts of areas and volumes. In a geometry lesson, we compare two figures and come across congruence, just as we read about the equality of numbers in an elementary math class.1

What is a Triangle?

One of the first shapes we learn is the triangle. A triangle is a two-dimensional shape with three sides, three angles, and three vertices.2 Triangles are three-sided closed figures that can be classified into various types according to their sides and angles. Equilateral triangles, isosceles triangles, and scalene triangles are the common variants.

So, what then, is a congruent triangle exactly?

Congruent triangles are those whose three sides and three angles match those of another triangle’s corresponding sides and angles. In other words, we can say that congruent triangles have equal corresponding sides and angles. To make the triangles appear identical, they can be rotated, flipped, turned, or slid. The triangles coincide with one another and can be superimposed on one another if they are repositioned.3 

“The symbol for congruent is ≅. Two triangles are congruent when the three sides and the three angles of one triangle have the same measurements as three sides and three angles of another triangle.”4

Congruent triangles
Congruent triangles

Applications of Triangle Congruence in the Real World

Engineering and Architecture

Congruent triangles are employed in the building process to strengthen the framework. This guarantees that the structures are rigid and strong. As a result, they are unaffected by strong winds or other weather-related conditions. Large man-made structures cannot be built without them. Considering a triangle is one of the most stable shapes and congruence is essential for producing even surfaces, this is the case. In geometric art, carpet designs, stepping stone patterns, and architectural designs, congruent triangles are also often used.3 

The two most prevalent examples of this could be:

  • Truss Bridge: Equilateral triangles are used to create truss bridges on both sides. All these triangles all meet the SSS congruence criteria, which states that if the length of three sides of one triangle equals the length of three sides of another triangle, then the two triangles are said to be congruent. This is due to the fact that the truss bridge requires equal weight-controlling lengths to maintain the structure and prevent collapse.3
Truss bridge
  • Geodesic Dome: This is common playground equipment. Angles and side lengths must be created such that all triangles involved are congruent under the SAS rule. The SAS rule says that if two sides of a triangle and the angle between them are equal to two sides and the angle of another triangle, then the two triangles are congruent. If one is not careful, one mistake in terms of angle, side length, or congruence can be dangerous to the point where it could be fatal.3

The Legend

Did you know that there is a well-known legend surrounding the use of congruent triangles? Legend has it that one of Napoleon’s officers calculated the width of a river using congruent triangles. The officer lowered the visor of his cap until the edge of the opposing bank was the furthest thing in his field of vision as he stood upright on the riverbank. After that, he turned and took note of the location on his side of the river that was parallel to his eye and the visor’s tip. The officer then measured the distance to this point and declared it to be the river’s width. If you want to confirm that he was right, you can use congruent triangles.5

Congruence is an intriguing idea that helps us develop and strengthen our mathematical reasoning abilities. It can serve as a measurement of perfection at times and be an important component of quality assurance. The path to understanding complex geometry concepts becomes a little easier after this shape-learning milestone is reached.1 After reading the examples above, if you feel inspired to search for congruence in your surroundings, please do so. You could also read about more interesting shapes, like radial shapes in nature and symmetry in space. You can also visit BYJU’S FutureSchool Blog to read more such fascinating articles.


  1. 20 Real-Life Examples Of Congruent Figures – Number Dyslexia. (n.d.). Retrieved July 5, 2022, from https://numberdyslexia.com/real-life-examples-of-congruent-figures/ 
  2. Congruent Triangles – Explanation & Examples. (n.d.). Retrieved July 5, 2022, from https://www.storyofmathematics.com/congruent-triangles/ 
  3. Top 3 Real Life Applications of Congruent Triangles. (n.d.). Retrieved July 5, 2022, from https://icrowdnewswire.com/2021/06/14/top-3-real-life-applications-of-congruent-triangles/ 
  4. Congruent Triangles. (n.d.). Retrieved July 5, 2022, from https://www.cliffsnotes.com/study-guides/geometry/triangles/congruent-triangles 
  5. Real World Applications – Understand congruence and similarity using physical models. (n.d.). Retrieved July 5, 2022, from https://understandingcongruences.weebly.com/real-world-applications.html