Triangles, being the focus of Trigonometry, deserve their own full section. Focus here is on types of triangles, the Pythagorean Theorem, the special triangles, and the area of a triangle (not requiring trigonometric functions). Many exercises involving the \( 30^{ \circ } \)-\( 60^{ \circ } \)-\( 90^{ \circ } \) and \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) that would normally be presented in Right Triangle Trigonometry are found in this section and its homework (but without reference to the trigonometric functions). Similar triangles are held off until next section.
Prerequisite Skills
Sets and Numbers
Inequalities and Inequality Notation
Geometry
Common Volumes
Simplifying Expressions
Properties of Radicals and Simplifying Expressions Involving Roots
Solving Equations
Solving Quadratic Equations (by Extraction of Roots)
Solving Radical Equations
Solving Inequalities
Solving Linear Inequalities
Solving Compound Inequalities
Triangle Basics
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 \( 180^{\circ} \).
The Triangle Inequality
Theorem: Triangle Inequality
In any triangle, we must have that\[p + q > r, \nonumber \]where \(p, q\), and \(r\) are the lengths of the sides of the triangle (see the figure below).
That is, the sum of the lengths of any two sides of a triangle is greater than the length of the other side.
Types of Triangles
Equilateral Triangles
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
All the angles of an equilateral triangle are equal.
Isosceles Triangles
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.
Theorem: Angles of an Isosceles Triangle
The base angles of an isosceles triangle are equal.
We introduce another common notation used throughout Geometry - the tick mark. We use tick marks on line segments to denote they have equal length. Similarly, tick marks on angles denote that the angles have the same measure.
Scalene Triangles
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.
Classification of Triangles by Largest Angle
Obtuse Triangles
Definition: Obtuse Triangle
A triangle which contains an obtuse angle is called an obtuse triangle.
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 \( 90^{ \circ } \).
Right Triangles
Definition: Right Triangle
A triangle containing a right angle is called a right triangle.
Right Triangles: The Pythagorean Theorem
Theorem: Pythagorean Theorem
In a right triangle, if \(c\) is the length of the hypotenuse, and the lengths of the two legs are denoted by \(a\) and \(b\) (as shown in the figure below), then\[a^2 + b^2 = c^2. \nonumber \]
Caution
The Pythagorean Theorem is valid only for right triangles!
Acute Triangles
Definition: Acute Triangle
A triangle in which all angles are less than \( 90^{ \circ } \) is called an acute triangle.
Special Triangles
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.
A right triangle in which one angle is \( 30^{ \circ } \) and another angle is \( 60^{ \circ } \) is called a \( 30^{ \circ } \)-\( 60^{ \circ } \)-\( 90^{ \circ } \) triangle.
Theorem: Side Relationships for a \( 30^{ \circ } \)-\( 60^{ \circ } \)-\( 90^{ \circ } \) Triangle
In any right triangle in which the two acute angles are \( 30^{ \circ } \) and \( 60^{ \circ } \), the hypotenuse is always twice the length of the shortest side (the side opposite the \( 30^{ \circ } \) angle), and the remaining side (opposite the \( 60^{ \circ } \) angle) is always \( \sqrt{3} \) times the shortest side.
Proof
Consider an equilateral triangle with side lengths \( 2a \), as shown in the figure below.
Since this triangle is equilateral, each angle is \( 60^{ \circ } \). Drawing an altitude divides the triangle into two smaller triangles. Since the altitude is perpendicular to the base, each of these smaller triangles is a right triangle. Moreover, since one angle in each of these smaller right triangles is \( 60^{ \circ } \), the remaining angle must be \( 30^{ \circ } \). Furthermore, the side opposite the \( 30^{ \circ } \) angle must have length \( a \) (in the original equilateral triangle, the side opposite \( 60^{ \circ } \) had length \( 2a \), so it should make sense that the side opposite \( 30^{ \circ } \) is half of this). We now consider one of these smaller right triangles (see the figure below).
The missing side length, which we will call \( x \) for now, can be found using the Pythagorean Theorem.\[\begin{array}{rrcl}
& x^2 + a^2 & = & (2a)^2 \\
\implies & x^2 + a^2 & = & 4a^2 \\
\implies & x^2 & = & 3a^2 \\
\implies & x & = & a \sqrt{3} \\
\end{array} \nonumber \]
Since an isosceles right triangle has two angles that are \( 45^{ \circ } \), it is commonly called a \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) triangle.
Theorem: Side Relationship for a \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) Triangle
In any right triangle in which the two acute angles are \( 45^{ \circ } \), the hypotenuse is always \( \sqrt{2} \) times the side length.
Proof
The sides opposite the \( 45^{ \circ } \) angles in a \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) triangle must have the same length. Let \( a \) be the lengths of these sides and \( x \) be the length of the hypotenuse, as shown in the figure below.
Since this is a right triangle, we can use the Pythagorean Theorem to find the length of the hypotenuse in terms of \( a \).\[\begin{array}{rrcl}
& a^2 + a^2 & = & x^2 \\
\implies & 2 a^2 & = & x^2 \\
\implies & a \sqrt{2} & = & x \\
\end{array} \nonumber \]
Area of a Triangle
Theorem: Area of a Triangle
Let \( b \) be the length of one side of a triangle and \( h \) be the altitude height from that side to its opposite vertex. Then the area of the triangle is given by\[ A = \dfrac{1}{2} bh. \nonumber \]
