Triangles may be classified according to the relative lengths of their sides:
An equilateral triangle has three equal sides,
An isosceles triangle has two equal sides.
A scalene triangle has no equal sides.
Triangles may also be classified according to the measure of their angles:
An acute triangle is a triangle with three acute angles.
An obtuse triangle is a triangle with one obtuse angle.
An equiangular triangle is a triangle with three equal angles,
Each angle of an equiangular triangle must be \(60^{\circ}\), We will show in section 2, 5 that equiangular triangle are the same as equilateral triangles,
A right triangle is a triangle with one right angle, The sides of the right angle are called the legs of the triangle and the remaining side is called the hypotenuse,
Example \(\PageIndex{1}\)
Find \(x\) if \(\triangle ABC\) is isosceles with \(AC = BC\):
An altitude of a triangle is a line segment from a vertex perpendicular to the opposite·side, In Figure 4, \(CD\) and \(GH\) are altitudes, Note that altitude \(GH\) lies outside \(\triangle EFG\) and side \(EF\) must be extended to meet it.
A median of a triangle is a line segment from a vertex to the midpoint of the opposite side, In Figure 5, CD is a median,
An angle bisector is a ray which divides an angle into two eaual angles. In Figure \(\PageIndex{6}\), \(\overrightarrow{CD}\) is an angle bisector.
We reject the answer \(x = 0\) because the measures of \(\angle ACD\) and \(\angle BCD\) must be greater than \(0^{\circ}\). Therefore \(\angle ACB = \angle ACD + \angle BCD = 30^{\circ} + 30^{\circ} = 60^{\circ}\).
Answer: \(\angle ACB = 60^{\circ}\).
The perimeter of a triangle is the sum of the lengths of the sides. The perimeter of \(\triangle ABC\) in Figure \(\PageIndex{7}\) is \(3 + 4 + 5 = 12\).
Theorem \(\PageIndex{1}\)
The sum of any two sides of a triangle is greater than the remaining side.
For example, in Figure \(\PageIndex{7}\), \(AC + BC = 3 + 4 > AB = 5\).
Proof
This follows from the postulate that the shortest distance between two noints is along a straight line, For example, in Figure \(\PageIndex{7}\), the length \(AB\) (a straight line segment) must be less than the combined lengths of \(AC\) and \(CB\) (not on a straight line from \(A\) to \(B\)),
Example \(\PageIndex{5}\)
Find the perimeter of the triangle in terms of \(x\), Then find the perimeter if \(x = 1\):