Triangles

⏱ 8 min✏️ Quiz at the end

Types of Triangles

A triangle is a polygon with three sides and three angles. Triangles are classified by their sides and by their angles.

Classification by Sides

Equilateral triangle:

  • All 3 sides equal in length
  • All 3 angles equal (each = 60°)
  • 3 lines of symmetry; rotational symmetry of order 3

Isosceles triangle:

  • Exactly 2 sides equal
  • The 2 base angles (opposite the equal sides) are equal
  • 1 line of symmetry

Scalene triangle:

  • No sides equal
  • No angles equal
  • No lines of symmetry

Classification by Angles

Acute triangle: all three angles are less than 90°

Right-angled triangle: one angle is exactly 90°; the side opposite the right angle is the hypotenuse (longest side)

Obtuse triangle: one angle is greater than 90°

Note: a triangle can belong to both categories, e.g. a right-angled isosceles triangle.

Angle Properties

Angles in a triangle always sum to 180°

This is one of the most useful facts in geometry. To find a missing angle: subtract the known angles from 180°.

Example: Angles A = 50°, B = 70°. Find C. C = 180° − 50° − 70° = 60°

Exterior angle theorem: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

If the interior angles are 50°, 70°, and 60°, the exterior angle at the 60° vertex = 50° + 70° = 120° (and 120° + 60° = 180° ✓, as they are on a straight line).

The Triangle Inequality

Any side of a triangle must be less than the sum of the other two sides. If this condition is not met, the three lengths cannot form a triangle.

Example: Can 3, 4, and 8 form a triangle? 3 + 4 = 7, which is less than 8 → Not a valid triangle

Example: Can 5, 7, and 10 form a triangle? 5 + 7 = 12 > 10; 5 + 10 = 15 > 7; 7 + 10 = 17 > 5 → Valid triangle

Area of a Triangle

Area = (1/2) × base × height

The height must be perpendicular to the base — it is not the slant side.

Example: Base = 8cm, perpendicular height = 5cm Area = (1/2) × 8 × 5 = 20cm²

For any triangle where two sides and the included angle are known: Area = (1/2) × a × b × sin(C)

Congruent Triangles

Two triangles are congruent if they are identical in shape and size — one can be placed exactly on top of the other (by flipping, rotating, or sliding).

Four conditions prove congruence:

ConditionMeaning
SSSAll three sides are equal
SASTwo sides and the included angle are equal
ASATwo angles and the included side are equal
RHSRight angle, hypotenuse, and one other side are equal

Note: AAA (all angles equal) does NOT prove congruence — it only proves similarity.

Example: Two triangles both have sides 5cm, 7cm with an included angle of 40°. By SAS, they are congruent.

Similar Triangles

Two triangles are similar if their corresponding angles are all equal. Similar triangles have the same shape but may be different sizes.

Properties of similar triangles:

  • Corresponding angles are equal
  • Corresponding sides are in the same ratio (scale factor)

If the scale factor between two similar triangles is k:

  • Corresponding lengths are in ratio k : 1
  • Areas are in ratio k² : 1

Example: Two similar triangles have sides in ratio 1:3. If the smaller has area 10cm², the larger has area 10 × 3² = 10 × 9 = 90cm²

Proving Triangles Similar

Show that two pairs of corresponding angles are equal (the third pair is then automatically equal since angles sum to 180°). This is the AA condition for similarity.

Pythagoras and Right-Angled Triangles

In a right-angled triangle: a² + b² = c², where c is the hypotenuse.

This is used to find a missing side when the other two sides are known (see the Pythagoras topic for full coverage).

Common Mistakes

  • Forgetting that the triangle inequality must hold — three lengths do not automatically form a triangle
  • Using AAA as a congruence condition (it only proves similarity)
  • Confusing "included" angle in SAS — it must be the angle between the two given sides
  • Using the slant height instead of the perpendicular height when calculating area

Tips and Tricks

  • When angles in a triangle are needed: angles sum to 180°, so find the missing one by subtraction
  • Isosceles triangles have two equal angles — if you know one base angle, the other is the same
  • To identify similar triangles in a diagram, look for parallel lines (which create equal corresponding or alternate angles)
  • Area = (1/2) × base × height always uses the perpendicular height — draw it in if it is not shown
  • The exterior angle shortcut is faster than going via the straight-line rule when working with multiple triangles
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