Parts of a triangle
A triangle ABC has three sides (AB, BC, CA), three angles (∠A, ∠B, ∠C) and three vertices (A, B, C).
Triangles are classified in two ways:
By sides | By angles |
|---|---|
Equilateral: all sides equal | Acute: all angles less than 90° |
Isosceles: two sides equal | Right: one angle is exactly 90° |
Scalene: no sides equal | Obtuse: one angle is more than 90° |
Medians and altitudes
A median joins a vertex to the midpoint of the opposite side. Every triangle has three medians, and they always meet at a point inside the triangle.
An altitude is the perpendicular line from a vertex to the opposite side (or to that side extended). Every triangle has three altitudes. In an obtuse triangle, some altitudes fall outside the triangle.
Angle sum property
The three angles of a triangle always add up to 180°.
You can check this yourself. Cut out a paper triangle, tear off its three corners, and place them side by side. They form a straight line, which is 180°.
Example. In △PQR, ∠P = 50° and ∠Q = 65°. Find ∠R.
∠R = 180° − (50° + 65°) = 65°.
Two consequences:
An equilateral triangle has three equal angles, so each is 180° ÷ 3 = 60°.
A triangle can have at most one right angle or one obtuse angle.
Exterior angle property
Extend one side of a triangle. The angle formed outside the triangle is an exterior angle.
An exterior angle of a triangle equals the sum of the two interior opposite angles.
Example. In △ABC, side BC is extended to D. If ∠A = 40° and ∠B = 75°, then the exterior angle ∠ACD = 40° + 75° = 115°.
This follows from the angle sum property: ∠ACB = 180° − 115° = 65°, and ∠ACB together with ∠ACD forms a straight line.
Isosceles triangles
In an isosceles triangle, the angles opposite the equal sides are also equal.
Example. An isosceles triangle has a vertex angle of 40°. The two base angles are equal, so each is (180° − 40°) ÷ 2 = 70°.
Sum of two sides
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
This is why you cannot make a triangle from sticks of 2 cm, 3 cm and 6 cm: 2 + 3 = 5 is less than 6, so the two short sticks cannot meet.
To test three lengths quickly, check that the two shorter sides add up to more than the longest side.
The Pythagoras property
In a right-angled triangle, the side opposite the right angle is the hypotenuse. It is the longest side.
(hypotenuse)² = (one leg)² + (other leg)²
Example. The legs of a right triangle are 6 cm and 8 cm. Find the hypotenuse.
h² = 6² + 8² = 36 + 64 = 100, so h = 10 cm.
The converse is also true: if a triangle's sides satisfy a² + b² = c², then the triangle has a right angle. So 5, 12, 13 forms a right triangle, because 25 + 144 = 169.
Example. A 15 m ladder leans against a wall, with its foot 9 m from the wall. How high up the wall does it reach?
Height² = 15² − 9² = 225 − 81 = 144, so the height is 12 m.