## Triangles

Triangles are polygons with three sides.

### Definitions

• vertex of a triangle is the intersection of two sides.
• median is a line from a vertex to the mid-point of the opposite side.
• An altitude is a line from a vertex that meets the opposite side at an angle of 90°.
• Two triangles are congruent if they have exactly the same size and shape.

### Types of Triangles

• An acute-angled triangle contains no angles greater than 90°.

e.g. A, B and C are all acute angles. • A right − angled triangle contains an angle of 90°.

e.g. A = 90 • An obtuse − angled triangle contains an angle between 90°and 180°. • A scalene triangle contains three sides unequal in length. All angle sizes are different.
• An isosceles triangle is one in which two of the sides and their opposite angles are equal.

Symmetry of Triangles

An isosceles triangle has one axis of symmetry. axis of symmetry

e.g. sides: a = b

angles: A = B

An equilateral triangle is one in which all three sides are of equal length and each interior angle is 60°.

An equilateral traingle has three axes of symmetry. e.g. sides: a = b = c

angles: A = B = C

### Properties of Triangles

• The interior angles of a triangle always add up to 180°.
• An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
• The exterior angles of a triangle always add up to 360°.
• The smallest angle of a triangle is opposite the smallest side.
• The largest angle of a triangle is opposite the largest side.

w = x + y 