Triangles • Topic 2 of 3

Properties of a Triangle—Angles and Sides

What are the Properties of a Triangle's Angles and Sides?

Triangles have beautiful internal mathematical relationships connecting their sides and angles. When you change the length of a side, the angle opposite to it changes too.

Here are the fundamental rules governing these properties:

  1. Isosceles Triangle Property: An isosceles triangle is a triangle that has two equal sides. The rule states that the angles opposite to the equal sides of an isosceles triangle are also equal. For example, if side AB = side AC in Triangle ABC, then Angle B (opposite side AC) must equal Angle C (opposite side AB).
  2. Converse of Isosceles Property: If two angles of a triangle are equal, then the sides opposite to those equal angles must also be equal in length.
  3. Equilateral Triangle Property: An equilateral triangle has all three sides equal. Because all sides are equal, all three internal angles must also be equal to one another. Since the total sum of angles in a triangle is always 180°, each angle of an equilateral triangle measures exactly 60° (180° divided by 3).

Let us check how these properties look:

Triangle TypeSide ConditionAngle ConditionSpecial Value
**Equilateral**All three sides are equalAll three angles are equalEach angle is always 60°
**Scalene**No sides are equalNo angles are equalAll angles are different
Important Triangle TheoremsIsoceles TriangleIf AB = AC, then ∠B = ∠C(angles opposite equal sides are equal)Angle-Side RelationGreater angle is oppositethe longer side (and vice versa)Triangle InequalitySum of any 2 sides > third sidea+b > c, b+c > a, a+c > bExterior AngleExterior angle = sum of thetwo non-adjacent interior anglesConverse of isoceles theorem: If ∠B = ∠C, then AB = ACAlways state the theorem and its converse separately
Solved Examples
1
Worked Example

Solve a standard problem on Properties of a Triangle—Angles and Sides.

Solution

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

Key Points

  • Understand the definition and properties of Properties of a Triangle—Angles and Sides.
  • 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.In an isosceles triangle, the base angles are:
Explanation: Equal.
Q2.The angle opposite the longest side is the:
Explanation: Largest.
Q3.The sum of any two sides of a triangle is ___ the third side.
Explanation: Greater than.
Q4.In an equilateral triangle, all angles are:
Explanation: Each $60^\circ$.
Practice more MCQs → 📝 Assignment · 35 marks