Class 7 Notes - Congruence of Triangles

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Congruence of Triangles

Congruence means "exactly equal in shape and size." Two figures are congruent if they have the same shape and size, even if they are flipped or rotated. In this chapter, we focus on the congruence of triangles.

What is Congruence?

  • Two line segments are congruent if they have the same length.
  • Two angles are congruent if they have the same measure.
  • Two triangles are congruent if their corresponding sides and angles are equal.

Congruence of Triangles

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

Criteria for Congruence of Triangles

  • SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
  • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
  • ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
  • AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
  • RHS (Right angle-Hypotenuse-Side): If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and one side of another right-angled triangle, the triangles are congruent.

Symbol for Congruence

The symbol for congruence is ≅ (three lines). For example, if triangle ABC is congruent to triangle DEF, we write:
△ABC ≅ △DEF

Examples

  • If AB = DE, BC = EF, CA = FD, then △ABC ≅ △DEF by SSS criterion.
  • If ∠A = ∠D, ∠B = ∠E, AB = DE, then △ABC ≅ △DEF by ASA criterion.

Practice Problems

  1. State whether the following pairs of triangles are congruent. If yes, mention the criterion used.
  2. Draw two triangles with sides 5 cm, 6 cm, and 7 cm. Are they congruent?
  3. In △PQR and △XYZ, PQ = XY, QR = YZ, ∠Q = ∠Y. Are the triangles congruent? Which criterion applies?

Summary

  • Congruent triangles have equal corresponding sides and angles.
  • SSS, SAS, ASA, AAS, and RHS are the main criteria for triangle congruence.
  • Congruence is denoted by the symbol ≅.