Chapter 7: Congruence of Triangles

 

Introduction to Congruence of Triangles

In mathematics, the concept of Congruence of Triangles refers to objects that are identical in shape and size. When we talk about triangles, congruence means that two triangles are exactly the same—they have equal sides and equal angles. This chapter introduces the idea of congruence, how to identify congruent triangles, and the criteria used to prove that two triangles are congruent.

1. What is Congruence?

• Two figures are said to be congruent if they can be placed one over the other and they fit perfectly without any gaps or overlaps.
• For triangles, congruence means:
  • All three sides of one triangle are equal to the three sides of the other triangle.
  • All three angles of one triangle are equal to the three angles of the other triangle.
• The symbol for congruence is ≅. For example, if triangle ABC is congruent to triangle DEF, we write ΔABC ≅ ΔDEF.

2. Congruence of Plane Figures

• Congruence is not limited to triangles; it applies to all plane figures (like squares, circles, etc.).
• For example, two squares are congruent if their sides are equal, and two circles are congruent if their radii are equal.
• However, this chapter focuses specifically on triangles.

3. Congruence of Triangles

• A triangle has three sides and three angles. For two triangles to be congruent:
  • The lengths of their corresponding sides must be equal.
  • The measures of their corresponding angles must be equal.
• Corresponding parts: When two triangles are congruent, their matching sides and angles are called corresponding sides and corresponding angles.

4. Notation for Congruent Triangles

• When we say ΔABC ≅ ΔDEF, it means:
  • Side AB = Side DE
  • Side BC = Side EF
  • Side CA = Side FD
  • ∠A = ∠D
  • ∠B = ∠E
  • ∠C = ∠F
• The order of vertices matters—it shows which sides and angles correspond.

5. Criteria for Congruence of Triangles

We don’t always need to measure all three sides and all three angles to prove two triangles are congruent. There are specific criteria (rules) that allow us to determine congruence with fewer measurements. These are:

(i) Side-Side-Side (SSS) Criterion

• If all three sides of one triangle are equal to the three sides of another triangle, the two triangles are congruent.
• Example: If in ΔABC and ΔDEF:
  • AB = DE
  • BC = EF
  • CA = FD
  • Then, ΔABC ≅ ΔDEF.

(ii) Side-Angle-Side (SAS) Criterion

• If two sides and the angle between them (included angle) in one triangle are equal to two sides and the included angle in another triangle, the triangles are congruent.
• Example: If in ΔABC and ΔDEF:
  • AB = DE
  • AC = DF
  • ∠A = ∠D (angle between AB and AC equals angle between DE and DF)
  • Then, ΔABC ≅ ΔDEF.

(iii) Angle-Side-Angle (ASA) Criterion

• If two angles and the side between them (included side) in one triangle are equal to two angles and the included side in another triangle, the triangles are congruent.
• Example: If in ΔABC and ΔDEF:
  • ∠A = ∠D
  • ∠B = ∠E
  • AB = DE (side between ∠A and ∠B equals side between ∠D and ∠E)
  • Then, ΔABC ≅ ΔDEF.

(iv) Right Angle-Hypotenuse-Side (RHS) Criterion

• This applies to right-angled triangles only. 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.
• Example: If in right-angled ΔABC (∠B = 90°) and ΔDEF (∠E = 90°):
  • AC = DF (hypotenuse)
  • AB = DE (one side)
  • Then, ΔABC ≅ ΔDEF.

6. Non-Criteria for Congruence

• AAA (Angle-Angle-Angle): If all three angles of one triangle equal all three angles of another, the triangles are similar, not necessarily congruent. Congruence requires equal sizes too.
• SSA (Side-Side-Angle): Two sides and a non-included angle being equal does not guarantee congruence, as it can result in two different triangles.

7. Properties of Congruent Triangles

• CPCT: Corresponding Parts of Congruent Triangles are equal. Once two triangles are proven congruent, all their corresponding sides and angles are equal.
• Congruent triangles have the same area, but the reverse is not true—triangles with the same area are not necessarily congruent.

8. Examples

Example 1 (SSS):

• Given: ΔPQR and ΔXYZ with PQ = XY = 5 cm, QR = YZ = 6 cm, PR = XZ = 7 cm.
• To Prove: ΔPQR ≅ ΔXYZ.
• Solution: Since all three sides are equal (PQ = XY, QR = YZ, PR = XZ), by the SSS criterion, ΔPQR ≅ ΔXYZ.

Example 2 (SAS):

• Given: In ΔABC and ΔDEF, AB = DE = 4 cm, AC = DF = 5 cm, ∠A = ∠D = 60°.
• To Prove: ΔABC ≅ ΔDEF.
• Solution: Two sides (AB = DE, AC = DF) and the included angle (∠A = ∠D) are equal. By the SAS criterion, ΔABC ≅ ΔDEF.

Example 3 (ASA):

• Given: In ΔLMN and ΔPQR, ∠L = ∠P = 50°, ∠M = ∠Q = 70°, LM = PQ = 3 cm.
• To Prove: ΔLMN ≅ ΔPQR.
• Solution: Two angles (∠L = ∠P, ∠M = ∠Q) and the included side (LM = PQ) are equal. By the ASA criterion, ΔLMN ≅ ΔPQR.

Example 4 (RHS):

• Given: In right-angled ΔABC (∠B = 90°) and ΔXYZ (∠Y = 90°), AC = XZ = 5 cm, BC = YZ = 3 cm.
• To Prove: ΔABC ≅ ΔXYZ.
• Solution: The hypotenuse (AC = XZ) and one side (BC = YZ) are equal in the right-angled triangles. By the RHS criterion, ΔABC ≅ ΔXYZ.

9. Key Points to Remember

• Congruence is about exact matches in size and shape.
• Use SSS, SAS, ASA, or RHS to prove congruence.
• AAA and SSA are not valid criteria for congruence.
• Always check the order of vertices when writing congruence statements.

10. Practical Applications

• Congruence is used in real life, such as in construction (to ensure identical parts), designing (to create symmetrical patterns), and even in solving puzzles or geometry problems.

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Chapter 8: Comparing Quantities

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