Geometry Study Guide: Classification and Properties of Triangles

Classification of Triangles by Interior Angle Size

Triangles are classified according to the measures of their interior angles (Podľa veľkosti uhlov). This fundamental classification includes three distinct types of triangles reviewed from the 6th-grade geometry curriculum (Opakovanie 6. ročník).

An acute-angled triangle (ostrouhlý trojuholník) is defined as a triangle in which all $3$ interior angles are acute (3 ostré uhly), meaning each angle measures strictly less than $90^{\text{circ}}$.

A right-angled triangle (pravouhlý trojuholník) contains exactly $1$ right angle measuring $90^{\text{circ}}$ and $2$ acute angles (1 pravý, 2 ostré uhly).

An obtuse-angled triangle (tupouhlý trojuholník) contains exactly $1$ obtuse angle measuring greater than $90^{\text{circ}}$ and $2$ acute angles (1 tupý, 2 ostré uhly).

Classification of Triangles by Side Lengths

Triangles are also categorized based on the relative lengths of their three sides (Podľa dĺžok strán).

A scalene triangle (rôznostranný trojuholník) has three sides of completely unequal lengths, where $a \neq b \neq c$ and $BC \neq AC \neq AB$.

An isosceles triangle (rovnoramenný trojuholník) has two sides of equal length.

An equilateral triangle (rovnostranný trojuholník) has all three sides of equal length.

Triangle Notation and Angle Sum Relationships

In standard geometric triangle notation, the three vertices of a triangle are labeled as $A$, $B$, and $C$. The sides opposite to these vertices are defined as $a = BC$, $b = AC$, and $c = AB$.

The interior angles at vertices $A$, $B$, and $C$ are denoted by Greek letters $\alpha$, $\beta$, and $\gamma$, respectively. The sum of all interior angles in any triangle is always equal to $180^{\circ}$:

$\alpha + \beta + \gamma = 180^{\circ}$

Exterior Angles and Angle Properties

Exterior angles of a triangle are designated as $\alpha'$, $\alpha''$, $\beta'$, $\beta''$, $\gamma'$, and $\gamma''$ (vonkajšie uhly).

An interior angle and its adjacent exterior angle (susedné uhly) form a linear pair on a straight line, which means their sum is equal to $180^{\circ}$:

$\alpha + \alpha' = 180^{\circ}$

Vertical angles (vrcholové uhly) formed at the same vertex are equal to each other:

$\alpha' = \alpha''$

Relationship Between Side Lengths and Opposite Angles

There is a strict ordering relationship between side lengths and opposite angle sizes in a triangle.

If a side is the shortest side in a triangle (for example, side $a$), then the interior angle directly opposite to that side (angle $\alpha$ or $\beta$) is the smallest interior angle.

If a side is the longest side in a triangle (for example, side $c$), then the interior angle directly opposite to that side (angle $\gamma$) is the largest interior angle.

These principles are essential for solving problems related to the perimeter (obvod) and area (obsah) of triangles.