In geometry, we have learned different types of shapes and sizes. Some of them are two-dimensional shapes and some are three-dimensional shapes. These shapes and sizes help us to understand and compare the shapes of different objects we have in real life. The example shapes are circle, triangle, square, rectangle, pentagon, hexagon, etc.
Two objects can be said to be similar if they have the same shape, but the size of both objects may vary. Two similar triangles are not always congruent to each other, but two congruent triangles are always similar. The criteria for congruency of triangles are different. Before we learn the similarity and congruency of triangles, let us understand what is the shape of triangles?
Shape of Triangle
One of the most important shapes that we have learned since our childhood is triangles. A triangle is a closed shape that has three sides and three vertices. Each of the vertex of the triangle form an angle with the two adjacent sides.
The triangles are further categorized into six different types, based on their sides and angles. They are:
- Scalene triangle (All the three sides are unequal)
- Isosceles triangle (Any two sides are equal)
- Equilateral triangle (All three sides are equal)
- Acute triangle (All the angles are less than 90 degrees)
- Obtuse triangle (Any of the angle is more than 90 degrees)
- Right triangle (One angle is equal to 90 degrees)
These triangles are the basic types of triangles. But if we compare any two triangles, then the similarity and congruency of the triangles comes into picture.
Now if two triangles are having the same shape but different sizes, then they are said to be similar triangles. See the example below of similar triangles.
In the above figure, the yellow triangle and blue triangle are having the same shape. But the blue triangle is bigger than the yellow triangle, thus their sizes vary. Hence, both the triangles are similar triangles.
Hence, we can say, similar triangles have the same shape, corresponding angles are equal and ratio of corresponding sides are also equal.
Two triangles are congruent if the three sides and three angles of the triangles are equal. So, in this case, not only the shape of the triangle should be the same but also its size. Hence, the congruent meaning in Maths is referred to as two figures that have the same shape and sizes.
To prove if two triangles are congruent, we have certain criteria, such as:
- SSS (Side-Side-Side)
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle)
- AAS (Angle-Angle-Side)
- RHS (Right angle-Hypotenuse-Side)
If all the three sides of a triangle are equal to three sides of another triangle, then the two triangles are said to be congruent.
If any two sides and angle between them of a triangle are equal to the corresponding two sides and angle of another triangle, then the two triangles are congruent.
If two triangles have any two corresponding angles and the side included between the two angles, equal in measure, then they are congruent.
A triangle with two angles and a non-included side of a triangle is equal to the corresponding angles and sides of another triangle, then the triangles are said to be congruent.
If a side and hypotenuse of a right triangle are equal in length with the corresponding side and hypotenuse of another right triangle, then both triangles are congruent to each other.
By this article, we conclude that the similarity of triangles differ from congruency. Also, both similar triangles and congruent triangles have different conditions.