Triangles • Topic 1 of 3

Congruence of Triangles and Criteria for Congruence

What is Congruence of Triangles?

In simple English, congruence means identical in every single way. If two shapes have the exact same shape and the exact same size, they are said to be congruent.

Imagine two identical brand-new one-rupee coins placed directly on top of each other. They cover each other completely because their boundaries match perfectly. In geometry, when two triangles are congruent, it means that if you cut one out and place it over the other, it will cover it exactly.

When two triangles, Triangle ABC and Triangle PQR, are congruent, we write it using the special symbol "≅":

Triangle ABC ≅ Triangle PQR

This symbol combines "~" (same shape) and "=" (same size).

When two triangles are congruent, their corresponding parts match up perfectly. This gives us a very important rule called CPCT, which stands for Corresponding Parts of Congruent Triangles. This means:

  • Corresponding sides are equal.
  • Corresponding angles are equal.

Instead of measuring all three sides and all three angles every time, mathematicians discovered four main shortcut rules or criteria for congruence:

  1. SAS (Side-Angle-Side): Two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle. (Note: The angle must be trapped tightly between the two sides).
  2. ASA (Angle-Side-Angle): Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle.
  3. SSS (Side-Side-Side): Two triangles are congruent if all three sides of one triangle are equal to the corresponding three sides of the other triangle.
  4. RHS (Right angle-Hypotenuse-Side): Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle.

Let us look at a quick comparison table of these rules:

Congruence RuleFull NameWhat Must Be Equal?Position Rule
**ASA**Angle-Side-Angle2 Angles and 1 SideThe side must be *between* the two angles.
**SSS**Side-Side-Side3 SidesAll three matching pairs of sides must be equal.
**RHS**Right angle-Hypotenuse-Side1 Right Angle, Hypotenuse, 1 SideApplies only to right-angled triangles.
Triangle Congruence ConditionsSSSSide-Side-SideAll 3 sides equalSASSide-Angle-Side2 sides + included angleASAAngle-Side-Angle2 angles + included sideAASAngle-Angle-Side2 angles + non-included sideRHSRight-Hyp-SideRight angle + hyp + sideSymbol: △ABC ≅ △DEF means they are congruentCorresponding parts are equal — CPCT (Corresponding Parts of Congruent Triangles)
Solved Examples
1
Worked Example

Solve a standard problem on Congruence of Triangles and Criteria for Congruence.

Solution

Apply the formula/method shown in the concept section above.

Key Points

  • Understand the definition and properties of Congruence of Triangles and Criteria for Congruence.
  • Study the worked examples and practice similar problems.
  • Always verify your answer using the original conditions.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Congruent triangles have equal corresponding sides and:
Explanation: Sides and angles equal.
Q2.SAS stands for:
Explanation: Side-Angle-Side.
Q3.ASA congruence uses two angles and the:
Explanation: Included side.
Q4.RHS congruence applies to:
Explanation: Right triangles.
Practice more MCQs → 📝 Assignment · 35 marks