A triangle is one of the most fundamental figures in plane geometry, and it is classified along two independent axes: the relative lengths of its sides and the measures of its interior angles.
Definition: Triangle
When three line segments bound a portion of the plane, the resulting shape is called a triangle (also known as a three-sided polygon). The line segments are called the sides of the triangle. A point where two sides meet is called a vertex of the triangle. Finally, the angle formed in the triangle's interior where two sides meet is called an angle of the triangle.
Theorem: Triangle Sum
The sum of the interior angles in a triangle is \( 180^{\circ} \).
Proof
Given the triangle, \( \triangle ABC \), draw a line parallel to side \( \overline{BC} \) going through vertex \( A \) (see the figure below).
Since \( \angle q + \angle a + \angle p \) forms a straight angle,\[ \angle q + \angle a + \angle p = 180^{\circ}. \nonumber \]Moreover, the line segment \( \overline{AC} \) is a transversal for the parallel lines \( \overline{BC} \) and \( \overline{DE} \). Therefore, \( \angle c = \angle p \). By a similar argument, \( \angle b = \angle q \). Hence,\[ \angle b + \angle a + \angle c = 180^{\circ}. \nonumber \]Thus, the sum of the angles of a triangle is \( 180^{\circ} \).
Definition: Equilateral Triangle
A triangle in which all three sides have the same length is called an equilateral triangle.
Theorem: Angles of an Equilateral Triangle
A triangle is equilateral if and only if it is equiangular, in which case each interior angle measures \(60^{\circ}\).
Definition: Isosceles Triangle
A triangle in which at least two sides are of equal length is called an isosceles triangle. The angle between the equal sides is called the vertex angle, and the other two angles are called the base angles.
Caution: Two Conventions for "Isosceles"
Under the inclusive definition used here—at least two equal sides—every equilateral triangle is also isosceles. Some texts instead require exactly two equal sides, which excludes the equilateral case. When reading a problem, confirm which convention is in force.
Theorem: Angles of an Isosceles Triangle
The base angles of an isosceles triangle are equal.
Theorem: Isosceles Triangle Theorem
In a triangle, two sides are equal in length if and only if the angles opposite those sides are equal in measure.
Definition: Isosceles Right Triangle
An isosceles triangle in which the vertex angle is \( 90^{ \circ } \) is called an isosceles right triangle.
Definition: Scalene Triangle
A triangle in which none of the sides have equal length (and, therefore, none of the angles are equal) is called a scalene triangle.
Definition: Obtuse Triangle
An obtuse triangle is a triangle that has one interior angle measuring more than \(90^{\circ}\).
Theorem: Angles of an Obtuse Triangle
In an obtuse triangle, one angle must be greater than \( 90^{ \circ } \), and the remaining two angles must each be less than \( 90^{ \circ } \).
Definition: Right Triangle
A right triangle is a triangle that has one interior angle measuring exactly \(90^{\circ}\). The side opposite the right angle is the hypotenuse. The two remaining sides are the legs.
Definition: Acute Triangle
An acute triangle is a triangle whose three interior angles each measure less than \(90^{\circ}\).
Definition: Altitude of a Triangle
The altitude of a triangle is a line segment drawn from a vertex of the triangle perpendicular to the opposite side (or the line containing the opposite side).
Theorem: Altitudes and Right Angles
The altitude of a triangle represents the shortest distance from the vertex to the opposite side and forms a \( 90^{ \circ } \) angle with that side.
Classifying Along Both Axes
The two classification schemes are independent, so a triangle is commonly named by both at once—for example, a "right isosceles triangle" or an "acute scalene triangle." Because the interior angles sum to \(180^{\circ}\), a triangle can contain at most one right or obtuse angle.
Examples
Example \(\PageIndex{1}\): Classifying by Sides
Classify each triangle by its side lengths.
A triangle with sides of length \(7\), \(7\), and \(7\).
A triangle with sides of length \(5\), \(5\), and \(9\).
A triangle with sides of length \(6\), \(8\), and \(11\).
Solutions
All three sides are equal, so the triangle is equilateral.
Exactly two sides are equal, so the triangle is isosceles.
No two sides are equal, so the triangle is scalene.
Example \(\PageIndex{2}\): Classifying by Angles
Classify each triangle by its interior angle measures.
A triangle with angles \(55^{\circ}\), \(65^{\circ}\), and \(60^{\circ}\).
A triangle with angles \(90^{\circ}\), \(35^{\circ}\), and \(55^{\circ}\).
A triangle with angles \(110^{\circ}\), \(40^{\circ}\), and \(30^{\circ}\).
Solutions
Each angle is less than \(90^{\circ}\), so the triangle is acute.
One angle equals \(90^{\circ}\), so the triangle is right.
One angle exceeds \(90^{\circ}\), so the triangle is obtuse.
Example \(\PageIndex{3}\): Finding a Missing Angle
A triangle has two interior angles measuring \(35^{\circ}\) and \(95^{\circ}\). Find the third angle and classify the triangle by its angles.
Solution
By the Triangle Angle Sum Theorem, the three angles total \(180^{\circ}\), so the third angle measures\[180^{\circ} - 35^{\circ} - 95^{\circ} = 50^{\circ}.\nonumber\]Since the angle measuring \(95^{\circ}\) is greater than \(90^{\circ}\), the triangle is obtuse.
Example \(\PageIndex{4}\): Applying the Isosceles Triangle Theorem
The triangle \(\triangle ABC\) is isosceles, with its two equal sides meeting at vertex \(A\), and \(\angle A = 40^{\circ}\). Find \(\angle B\) and \(\angle C\).
Solution
The angles opposite the two equal sides are \(\angle B\) and \(\angle C\). By the Isosceles Triangle Theorem these angles are equal, so let each measure \(x\). By the Triangle Angle Sum Theorem,\[40^{\circ} + x + x = 180^{\circ},\nonumber\]hence \(2x = 140^{\circ}\) and \(x = 70^{\circ}\). Therefore \(\angle B = \angle C = 70^{\circ}\).
Sources
Several parts of this text use modifications from the following source:
