Congruency means two or more objects have exactly the same shape and size, or one is the mirror image of the other. It describes the relationship between geometric figures that can be placed over each other and match perfectly. In mathematics, shapes are congruent if their corresponding sides and angles are equal, so they can be superimposed side by side and angle to angle. Congruency applies to line segments, angles, and all geometric shapes.
Mathematically, congruency is represented by the symbol ≅. .
Significance of Congruency in Mathematics
Congruency holds vital importance in the mathematical field. Following are some applications of congruency:
- It helps in analyzing shapes and their properties.
- It is useful in calculating accurate measurements.
- It is used in various trigonometric solutions.
Congruency in Geometry
In geometry, congruency is used to identify similar shapes. Even if a shape is rotated or flipped, the congruent shapes remain the same.
By definition, two figures, objects, or shapes are said to be congruent if they have the same shape or size as the mirror image of the other. The congruent shapes are superimposed on each other. This simply means that they can be placed completely over each other.
For example, two squares with sides of 5 cm each are congruent to each other, as they can completely overlap each other.
Geometric Concepts Related to Congruency
To understand the relationship between shapes and make deductions about their properties, we need to identify congruency in any geometrical shape. Following are some geometrical concepts related to congruency:
- Line Segment: Two lines are congruent, and they will have the same properties if their length is the same.
- Angles: Two angles will be congruent if they have the same measure.
- Circle: Two circles are congruent if they have the same diameter.
Overall, in order to find congruency in any geometrical shape, we need to identify them using different parameters. For example, in the case of polygons, we first match the vertices and then move towards sides and angles to find similarity.
Identifying Congruent Figures
Congruent figures do have some differences, but in terms of properties, they tend to have the same measures, which help us in analyzing two shapes accurately. To identify congruent figures in any polygon, we can follow the following steps:
Step 1: Check the type of 2-dimensional shape (triangle, hexagon, etc.) by counting the number of vertices.
Step 2: Check the length of all sides of the figure.
Step 3: Check the measure of all angles of the figure.
If all the measures of the given figures are the same, then the given figures are congruent to each other.
Type of Congruency
Congruency can be of different types depending upon its geometrical shape. Congruency can be defined in line segments with the same length, angles with the same measure, triangles where sides and angles are equal, circles with the same diameter or radius, etc. Depending upon the identification of congruency, it is divided into three major types.
1. Reflectional Congruence
Reflectional congruence is a congruency where two figures remain the same after flipping through a line of reflection. This congruency involves a reflection, which is a type of transformation where a figure is flipped over a line. Here the two figures are identical after getting reflected across a line of reflection, which acts like a line of symmetry.
Two figures that are reflectionally congruent can be overlapped on each other after aligning corresponding vertexes. This property helps in identifying symmetry between two geometrical shapes.
2. Rotational Congruence
Rotational congruence refers to the congruency where two shapes become the same (congruent) when one is turned around a center point.
In simple terms, two figures are congruent rotationally if one of the figures can be transformed to the other figure after a specific number of rotations around a center of rotation. Here, one figure is matched with the other figure by rotating it by a specific angle and direction to make them identical or congruent.
3. Translational Congruence
Translational congruence involves a translation, where a figure is moved from one location to another without changing its orientation or shape. Here two geometric shapes are congruent if, when they are translated in a specific direction, both the figures can be overlapped on each other by moving the figure parallel to the direction of the translation.
One important thing in translational congruency is that the orientation remains the same; only the figure slides in a specific direction without rotating or flipping it.
Properties
Congruent figures are shapes that completely overlap each other with the same shape and size. Their properties depend on the type of figure, such as line segments, angles, triangles, or other polygons.
- Congruent Angles and Sides: Two angles are congruent if their measures are equal. Similarly, sides are congruent if their lengths are equal. Polygons like triangles and squares use equal corresponding sides and angles to prove congruency.
- Corresponding Parts of Congruent Figures (CPCT): When two figures are congruent, all their corresponding sides and angles are equal. If one figure is superimposed on the other, every matching part coincides exactly. For example, if △ABC ≅ △CDA, then corresponding sides and angles are equal accordingly.
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Solving Problems On Congruency
Example 1: ABCD is a square, where AC is a diagonal. Prove that triangles ABC and CDA are congruent.
ABCD is a square, so all four sides are congruent. By the properties of square, we also know each angle formed by square is 90°.
Hence we have,
AB = CD
BC = DA
Also, Both Triangles have one side common i.e. AC,
By SSS Congruence theorem. Both triangles, ABC and CDA are congruent.
Example 2: Check whether the given quadrilaterals are congruent or not.
In the figure ABCD,
∠ABC = 90
But In PQRS
∠PQR ≠ 90
Hence, the angles of first quadrilateral are not equal to the angles of second quadrilateral. Therefore, the above given figures are not congruent to each other.