Criteria for Congruence of Triangles

Congruence of triangles
Consider triangles ABC and XYZ. Cut triangle ABC and place it over XYZ. The two triangles cover each other exactly, and they are of the same shape and size. Also notice that A falls on X, B on Y, and C on Z. Also, side AB falls along XY, side BC along YZ, and side AC along XZ. So, we can say that triangle ABC is congruent to triangle XYZ. Symbolically, it is represented as ΔABC ≅ ΔXYZ.

So, in general, we can say that two triangles are congruent if all the sides and all the angles of one triangle are equal to the corresponding sides and angles of the other triangle.

Order of the letters in the names of the congruent triangles shows the corresponding relationship. In two congruent triangles ABC and XYZ, the corresponding vertices are A and X, B and Y, and C and Z, that is, A corresponds to X, B to Y, and C to Z. Similarly, the corresponding sides are AB and XY, BC and YZ, and AC and XZ. Also, angle A corresponds to X, B to Y, and C to Z. So, we write ABC corresponds to XYZ.


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Criteria for congruence of triangles
Two triangles can be congruent if three of their corresponding parts are equal.


SSS congruence criterion
Two triangles are congruent if three sides of one triangle are equal to the three corresponding sides of the other triangle.

SAS congruence criterion
Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.

RHS congruence criterion
Two right-angled triangles are congruent if the hypotenuse and a side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle.

ASA congruence criterion
Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.

Two triangles with equal corresponding angles may not be congruent. So, there is no such thing as AAA congruence of triangles. Two congruent triangles have equal areas and equal perimeters.

Summary

Congruence of triangles
Consider triangles ABC and XYZ. Cut triangle ABC and place it over XYZ. The two triangles cover each other exactly, and they are of the same shape and size. Also notice that A falls on X, B on Y, and C on Z. Also, side AB falls along XY, side BC along YZ, and side AC along XZ. So, we can say that triangle ABC is congruent to triangle XYZ. Symbolically, it is represented as ΔABC ≅ ΔXYZ.

So, in general, we can say that two triangles are congruent if all the sides and all the angles of one triangle are equal to the corresponding sides and angles of the other triangle.

Order of the letters in the names of the congruent triangles shows the corresponding relationship. In two congruent triangles ABC and XYZ, the corresponding vertices are A and X, B and Y, and C and Z, that is, A corresponds to X, B to Y, and C to Z. Similarly, the corresponding sides are AB and XY, BC and YZ, and AC and XZ. Also, angle A corresponds to X, B to Y, and C to Z. So, we write ABC corresponds to XYZ.


triangles, triangle, congruent, sides, angles, same shape and size, corresponding sides and angles, two congruent triangles, corresponding sides, corresponding vertices, ABC corresponds to XYZ

triangles, triangle, congruent, sides, angles, same shape and size, corresponding sides and angles, two congruent triangles, corresponding sides, corresponding vertices, ABC corresponds to XYZ

triangles, triangle, congruent, sides, angles, same shape and size, corresponding sides and angles, two congruent triangles, corresponding sides, corresponding vertices, ABC corresponds to XYZ


Criteria for congruence of triangles
Two triangles can be congruent if three of their corresponding parts are equal.


SSS congruence criterion
Two triangles are congruent if three sides of one triangle are equal to the three corresponding sides of the other triangle.

SAS congruence criterion
Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.

RHS congruence criterion
Two right-angled triangles are congruent if the hypotenuse and a side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle.

ASA congruence criterion
Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.

Two triangles with equal corresponding angles may not be congruent. So, there is no such thing as AAA congruence of triangles. Two congruent triangles have equal areas and equal perimeters.

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