Math's: Congruence: Congruence of Triangles, and Criteria for Congruence of Triangles

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Congruence

Two figures, which have the same shape and same size are called congruent figures and this property is called congruence.

Congruent figures, when placed one over another, exactly coincide with one another or cover each other. In other words, two figures will be congruent, if parts of one figure are equal to the corresponding parts of the other.

  • Two-line segments are congruent if they are of equal length

  • Two polygons are equal if they have same number of sides and all of their sides are equal or their areas are equal

  • Two circles are congruent if their radii are equal.

Congruence of Triangles

Congruence of Triangles

Congruence of Triangles

Two triangles are congruent, if all the sides and all the angles of one are equal to the corresponding sides and angles of other.

Congruence of Triangles

Congruence of Triangles

  • For Example, in the above triangles, we can say, PQR is congruent to XYZ and we write

  • Congruence of two figures also implies that their corresponding (matching) parts are also equal. In the above triangles, P corresponds to X, Q corresponds to Y and R corresponds to Z. Also, and .

Criteria for Congruence of Triangles

In order to prove the congruence of two triangles, we are required to prove the equality of any of the three of their corresponding parts. Some of the Important criteria for proving the congruence of two triangles are-

  • SAS (Side- Angle- Side) Criteria- If any two sides and the included angle of one triangle are equal to the corresponding sides and the included angle of the other triangle, the two triangles are congruent.

  • In the following pair of triangles ABC and PQR, and. So, by SAS criteria.

Criteria for Congruence of Triangles

Criteria for Congruence of Triangles

  • ASA or AAS Criteria (Angle- Side- Angle or Angle- Angle- Side)- If any two angles and one side of a triangle are equal to corresponding angles and the side of another triangle, then the two triangles are congruent.

  • In the following pair of triangles ABC and PQR, and Thus, by ASA criteria

Criteria all the Three Sides

Criteria All the Three Sides

  • SSS (Side- Side- Side) Criteria- If all the three sides of one triangle are equal to the corresponding sides of another triangle, then the two triangles are congruent.

  • In the following pair of triangles ABC and PQR, and. Therefore, by SSS Criteria

Criteria the Hypotenuse

Criteria the Hypotenuse

  • RHS (Right Angle Hypotenuse Side) Criteria- If the hypotenuse and a side of one right triangle are respectively equal to the hypotenuse and a side of another right triangle, then the two triangles are congruent. It is to be noted here that one of the angles are already equal (90°) in two right triangles, so we need to prove the equality of two sides only.

  • In the following pair of right triangles ABC and PQR, . Therefore, by RHS Criteria

Criteria of Congruence

Criteria of Congruence

Criteria of Congruence

Criteria of Congruence

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