Properties of Triangles

• Angle sum property — sum of all interior angles of a triangle = 180°.
• Exterior angle property — exterior angle = sum of two non-adjacent interior angles.
• Triangle inequality — sum of any two sides > third side.
• Types: equilateral, isosceles, scalene; acute, right, obtuse.
• Pythagoras theorem (right triangle): a² + b² = c² where c is hypotenuse.

Level 1 — Angle sum property

In any triangle ABC: ∠A + ∠B + ∠C = 180°

If ∠A = 60°, ∠B = 70°, then ∠C = 180° − 60° − 70° = 50°.

Level 2 — Exterior angle property

An exterior angle of a triangle = sum of the two interior opposite angles.

If exterior angle at C = ∠D, then ∠D = ∠A + ∠B.

Level 3 — Triangle inequality theorem

For sides a, b, c to form a triangle:

• a + b > c
• b + c > a
• a + c > b

If any condition fails, the triangle cannot exist.

Level 4 — Pythagoras theorem (right triangle)

In a right-angled triangle: (hypotenuse)² = (base)² + (perpendicular)²

c² = a² + b²

Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17).

Triangle has angles in ratio 2 : 3 : 4. Find each angle.

Step 1 — Let angles = 2x, 3x, 4x.
Step 2 — Angle sum: 2x + 3x + 4x = 180° → 9x = 180° → x = 20°.
Step 3 — Angles: 2×20 = 40°, 3×20 = 60°, 4×20 = 80°.
Step 4 — Check: 40 + 60 + 80 = 180° ✓
Conclusion: angles are 40°, 60°, 80°. Triangle is acute scalene.

Right triangle with legs 9 cm and 12 cm. Find hypotenuse.

c² = 9² + 12² = 81 + 144 = 225
c = √225 = 15 cm

• Find the third angle if two angles are 55° and 75°.
• An exterior angle of a triangle is 110°. One interior opposite angle is 60°. Find the other.
• Can 4 cm, 5 cm, 10 cm form a triangle? Why?
• A right triangle has legs 6 m and 8 m. Find the hypotenuse.

Open the Practice tab for graded questions on Triangle Properties.