Triangles • Topic 2 of 3

Congruence of Triangles

What is Congruence? Two triangles are congruent if they have the same shape and same size. All corresponding sides are equal and all corresponding angles are equal.

Notation: \(\triangle ABC \cong \triangle PQR\) means:

  • \(AB = PQ\), \(BC = QR\), \(CA = RP\)
  • \(\angle A = \angle P\), \(\angle B = \angle Q\), \(\angle C = \angle R\)

Conditions for Congruence (SSS, SAS, ASA, RHS):

ConditionFull FormExplanation
SSSSide-Side-SideAll three sides of one triangle equal to all three sides of another
SASSide-Angle-SideTwo sides and the included angle are equal
ASAAngle-Side-AngleTwo angles and the included side are equal
AASAngle-Angle-SideTwo angles and a non-included side are equal
RHSRight-Hypotenuse-SideRight triangle with hypotenuse and one side equal

Important: AAA (Angle-Angle-Angle) is NOT a congruence condition (triangles can be similar but not congruent).

Triangle Congruence ConditionsSSS3 sides equalSAS2 sides + included angleASA2 angles + included sideAAS2 angles + non-included sideRHSRight angle + hypotenuse + sideCongruent triangles (△ABC ≅ △DEF) have:✓ All three sides equal ✓ All three angles equal ✓ Same areaNote: Order of vertices matters — △ABC ≅ △DEF means A↔D, B↔E, C↔FSymbol: ≅ (is congruent to) CPCT: Corresponding Parts of Congruent Triangles
Solved Examples
1
Worked Example
In \(\triangle ABC\) and \(\triangle DEF\), \(AB = DE\), \(BC = EF\), and \(AC = DF\). Are the triangles congruent? If yes, by which condition?
Solution- All three sides of one triangle equal all three sides of the other - This satisfies **SSS** (Side-Side-Side) condition - **Answer:** Yes, by SSS congruence *Example 2: In \(\triangle PQR\) and \(\triangle XYZ\), \(PQ = XY\), \(PR = XZ\), and \(\angle P = \angle X = 90°\). Are the triangles congruent? Solution: - Right triangles with hypotenuse? Need to check which are the hypotenuses - If \(PQ\) and \(XY\) are legs, and \(PR\) and \(XZ\) are legs, then \(QR\) and \(YZ\) are hypotenuses - But we only know two sides and the included right angle → **SAS** condition - **Answer:** Yes, by SAS congruence *Example 3: In \(\triangle ABC\) and \(\triangle DEF\), \(\angle A = \angle D = 50°\), \(\angle B = \angle E = 60°\), and \(AB = DE = 5\) cm. Are the triangles congruent? Solution: - Two angles are equal: \(\angle A = \angle D\), \(\angle B = \angle E\) - Third angles automatically equal: \(\angle C = \angle F = 70°\) - Included side \(AB = DE\) is between the equal angles - This satisfies **ASA** (Angle-Side-Angle) condition - **Answer:** Yes, by ASA congruence

Key Points

  • Congruent triangles have same shape and size
  • SSS: all three sides equal
  • SAS: two sides and included angle equal
  • ASA: two angles and included side equal
  • AAS: two angles and non-included side equal
  • RHS: right triangle with hypotenuse and one side equal
  • AAA is NOT a congruence condition
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Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Congruent triangles have the same:
Explanation: Identical shape and size.
Q2.SAS stands for:
Explanation: Side-Angle-Side.
Q3.ASA stands for:
Explanation: Angle-Side-Angle.
Q4.RHS applies to ___ triangles.
Explanation: Right-angled.
Practice more MCQs → 📝 Assignment · 35 marks