0.1: Triangles

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A triangle is formed when three straight line segments bound a portion of the plane. The line segments are called the sides of the triangle. A point where two sides meet is called a vertex of the triangle, and the angle formed is called an angle of the triangle. The symbol for triangle is $\triangle$.

The triangle in Figure $\PageIndex{1}$ is denoted by $\triangle ABC$ (or $\triangle BCA$ or $\triangle CAB$, etc.).

Its sides are $AB$, $AC$, and $BC$.
Its vertices are $A, B$, and $C$.
Its angles are $\angle A$, $\angle B$, and $\angle C$.

屏幕快照 2020-10-29 下午5.52.39.png — Figure $\PageIndex{1}$: Triangle $ABC$.

A fundamental property of triangles is the following:

Theorem $\PageIndex{1}$

The sum of the angles of a triangle is $180^{\circ}$.

In $\triangle ABC$ of Figure $\PageIndex{1}$, $\angle A + \angle B + \angle C = 180^{\circ}$.

Example $\PageIndex{1}$

Find $\angle C$:

$屏幕快照 2020-10-29 下午5.56.43.png$

Solution

\[\begin{array} {rcl} {\angle A + \angle B + \angle C} & = & {180^{\circ}} \\ {40^{\circ} + 60^{\circ} + \angle C} & = & {180^{\circ}} \\ {100^{\circ} + \angle C} & = & {180^{\circ}} \\ {\angle C} & = & {180^{\circ} - 100^{\circ}} \\ {\angle C} & = & {80^{\circ}} \end{array} \nonumber\]

Answer: $\angle C = 80^{\circ}$

Example $\PageIndex{2}$

Find $x$:

$屏幕快照 2020-10-29 下午6.57.45.png$

Solution

\[\begin{array} {rcl} {\angle A + \angle B + \angle C} & = & {180^{\circ}} \\ {2x + 3x + 4x} & = & {180} \\ {9x} & = & {180} \\ {x} & = & {20} \end{array} \nonumber\]

Check:

$屏幕快照 2020-10-29 下午7.00.53.png$

Answer: $x = 20$.

Classification of Triangles

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.

屏幕快照 2020-10-29 下午7.58.38.png — Figure $\PageIndex{1}$: Triangles classified according to their 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,

屏幕快照 2020-10-29 下午8.00.49.png — Figure $\PageIndex{2}$: Triangles classified according to their angles,

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,

屏幕快照 2020-10-29 下午8.02.49.png — Figure $\PageIndex{3}$: Right triangles.

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Solution

Solution

Theorem \(\PageIndex{1}\)

Example \(\PageIndex{1}\)

Solution

Example \(\PageIndex{2}\)

Solution

Classification of Triangles