Geometry - Congruent
- Introduction
- Congruent Definition
- What is Congruent in Geometry?
- Symbol of Congruent
- Congruent Line Segments
- Congruent Angles
- Congruent Triangles
- Congruent Circles
- Properties of Congruence in Geometry
- Application of Congruence in Geometry
- Conclusion
- Practice Problems
- Frequently Asked Questions
Introduction
The mathematical study of the features and relationships between shapes and figures is known as geometry. Congruence is one of the basic concepts of geometry. When two geometric objects or figures are same in size and shape, it is said that they are congruent. This article will discuss the concept of congruence in geometry as well as its properties and applications to various geometric problems.
Congruent Definition
Congruence, a fundamental concept in geometry, allows us to compare and group shapes according to differences and similarities. If two geometric shapes can be changed into one another by rigid motions like translations, rotations, and reflections without altering the shape or dimensions, they are said to be congruent.
What is Congruent in Geometry?
Two figures are said to be "congruent" if they can be positioned precisely over one another.
"Congruent" refers to things that are precisely the same size and shape. Even after the forms have been flipped, turned, or rotated, the shape and size ought to remain constant.
Example:
Symbol of Congruent
Congruence is symbolized by the symbol `≅`.
Since equal size and shape are implied by congruence in things, the symbol for congruence is made up of two symbols, one positioned above the other. The tilde ("`~`") symbol denotes resemblance in shape, while the equal sign ("`=`") denotes size equivalence.
Congruence is thus symbolized by the character `≅`.
If two things `A` and `B` are congruent, we will write them as:
`A ≅ B`
Congruent Line Segments
Congruence indicates equal shape and size, therefore if the line segments have the same shape and size, they will be congruent.
Examine the below image attentively.
`PQ` and `AB` share the same shape since they are both line segments. `PQ` and `AB` each measure five centimeters in length. As a result, the length of both line segments is the same. Two or more line segments are said to be congruent to one another if their lengths are equivalent. As a result, the line segments `AB` and `PQ` are congruent.
It will therefore be shown as:
Line segment `AB ≅` Line segment `PQ`
Congruent Angles
Angles with the same measures are said to be congruent angles. This indicates that their degree standards are comparable. Angles that are congruent are common in many geometric situations and are essential for building links between lines and objects.
`∠A` is congruent to `∠B`.
`∠A ≅ ∠B`
Congruent Triangles
Triangles that are equal in size and shape are known as congruent triangles. Two triangles are said to be congruent if their corresponding sides and angles are all equal.
`△ABC` is congruent to `△PRQ`.
`△ABC ≅ △PRQ`
Congruent Circles
Circles of the same size and shape are said to be congruent. As a result, their centers are situated at the same place in the plane, and their radii are equal. Essentially, two congruent circles would entirely overlap if they were superimposed on top of one another.
Properties of Congruence in Geometry
- Reflexive Property: Every geometric shape has the reflexive property of being congruent to itself. The geometry of congruence is based on this characteristic. In essence, it indicates that a shape is constantly congruent to itself, regardless of its location or orientation.
- Symmetric Property: Figures `A` and `B` are congruent if and only if they share the same symmetry. Congruence is a symmetric relation, therefore if one figure can be turned into another, the opposite transformation is also feasible.
- Transitive Property: The transitive property states that if figure `A` is congruent with figure `B` and figure `B` is congruent with figure `C`, then figure `A` is congruent with figure `C`. Through a series of connected figures, we can extend the idea of congruence.
Application of Congruence in Geometry
Triangle congruence: Congruence is a tool for demonstrating the equality of triangle sides and angles. Triangle congruence can be established using a variety of techniques, including SSS (Side-Side-Side), SAS (Side-Angle-Side), and ASA (Angle-Side-Angle).
Congruence of Polygons: Polygon congruence is a crucial factor in comparing and categorizing polygons. Congruent rectangles, for instance, have equal angles and side lengths.
Geometric Figure Construction: Congruence is used in geometric figure construction. We can utilize congruence to create an exact duplicate of a given geometric figure with a compass and straightedge.
Transformation Geometry: Congruence is essential to transformation geometry, which uses translations, rotations, and reflections to change the position of objects. It guarantees that the altered figure keeps its original size and shape.
Problem Solving: Congruence is a common tool for solving difficult geometric problems. Mathematicians can simplify issues and prove geometric shape-related theorems by recognizing congruent portions of figures.
Conclusion
A fundamental idea in geometry called congruence enables mathematicians to compare and analyze geometric figures in a methodical way. We may resolve issues, develop precise constructions, and establish theorems that serve as the foundation of geometry by comprehending the characteristics of congruent figures and using them in a variety of geometric contexts. The idea of congruence not only makes studying shapes and figures easier but also offers a strong framework for future investigation and discovery in the field of geometry.
Practice Problems
Q`1`. Congruent angles are angles that have the same:
- Measure
- Side
- Vertex
- Perimeter
Answer: a
Q`2`. If two rectangles have same length and breadth, they are:
- Equal
- Similar
- Congruent
- None of these
Answer: c
Q`3`. Find the incorrect response in the scenario with two congruent figures.
- Equal-sized figures are congruent.
- Congruent figures don't have the same shape.
- Rotating congruent figures is possible.
- None of these
Answer: b
Frequently Asked Questions
Q`1`. When can two figures be said to be congruent?
Answer: If two figures have the same size and shape, they are said to be congruent.
Q`2`. Can the sizes of congruent shapes vary?
Answer: The sizes of congruent figures cannot differ. Instead, if the figures are the same size and shape, they are said to be congruent. Similar figures are ones whose sizes are different but whose shapes are the same.
Q`3`. Are congruent figures all the same?
Answer: Yes, congruent figures are all identical.
Q`4`. Are circles that are congruent of the same diameter?
Answer: If the radius of two circles is the same, they are said to be congruent. The diameter is said to be twice the radius. Therefore, the diameter of congruent circles is the same.
Q`5`. Could stars be congruent?
Answer: Two stars are only deemed congruent if their sizes and shapes are identical regardless of how they are flipped, rotated, or superimposed upon one another.