Congruence and Similarity of Triangles

The measurements of the sides and angles of two or more triangles determine their congruence. A triangle’s size is determined by its three sides, and its shape is determined by its three angles. If the pairs of corresponding parts of a congruent triangle are equal, they are said to be congruent. In triangles, there are several conditions for congruence.

Congruent triangles

The following are the rules of the corresponding parts of a congruent triangle:

1. AA (Angle-Angle)

The triangles are comparable if two pairs of corresponding angles in a pair of triangles are congruent because if two angle pairs are equal, the third pair must be equal. The three pairs of sides must be proportionate when all three angle pairs are equal. Take into account the triangle angles that are hanging in the atmosphere. They don’t define the size of the triangle by themselves if they are the vertices of a triangle because they can move farther or closer to each other. The triangle they make stays the same shape when they move. As a result, they always produce similar triangles.

2. SSS (Side-Side-Side) 

Because we’re working with triangles, we know that if we’re given the same three sides, they’ll have the same three angles. As a result, two triangles that have the same sides are congruent. (It’s worth noting that two triangles with three equal angles don’t have to be congruent. All we know is that these triangles look alike.)

3. SAS (Side-Angle-Side)

The two triangles they make are comparable if two pairs of corresponding sides are in proportion and the included angle of each pair is equal. All three vertices of a triangle are fixed when two sides of the triangle and their included angle are fixed. The third pair of sides must be proportional because all three vertices are fixed, and two of the pairs of sides are proportional.

4. RHS (Right Angle-Hypotenuse-Side)

Right Angle-Hypotenuse-Side (RHS) is a congruence of triangles. The RHS congruence theorem asserts that if one right-angled triangle’s hypotenuse and side are equal to another right-angled triangle’s hypotenuse and matching side, the two triangles are congruent.

Only right-angled triangles are subject to this criterion.

It’s worth noting that if we maintain the hypotenuse and any of the other two sides of two right triangles are equal, we’ll automatically have three comparable sides because all three sides of a right triangle are related to each other. This is known as the Pythagoras theorem.

Conditions of congruence in triangles

If two triangles are of the same size and shape, they are said to be congruent. It is not necessary to find all six corresponding elements of both triangles to determine that they are congruent. According to studies and trials, there are five prerequisites for two triangles to be congruent. The congruence properties are SSS, SAS, ASA, AAS and RHS.

Congruence and similarity of triangles for SSC

When two triangles have the same shape and size, they are said to be congruent; their corresponding sides and angles are the same.

If two triangles share the same shape, they are considered to be comparable. In other words, if two triangles’ respective angles are equal and they are corresponding parts of a congruent triangle, they are similar.

Important notes on the congruence of triangles

• Two triangles are congruent if the six pieces of one triangle (three sides and three angles) are the same as the six parts of the other triangle 

• Five conditions must be met to establish whether two triangles are congruent. The criteria are SSS, SAS, ASA, AAS and RHS

• Because one triangle may be an enlarged replica of the other, two triangles with equal corresponding angles may not be congruent. As a result, there is no AAA congruence criteria

• The symbol (≅) is used to signify congruency

• If the perimeters and areas of two triangles are equal, they are said to be congruent

Conclusion

Triangles are classified as plane enclosed forms with three sides depending on their sides and angles. Isosceles, equilateral and scalene are some kinds of triangles. Even though the triangles have the same size and shape, one will appear to be a mirror image of the other. In simple terms, any object that seems to be the same figure when placed over its counterpart is congruent.