The Triangle and its Properties - Angle Sum Property of a Triangle

The Angle Sum Property states that the sum of the measures of the three interior angles of any triangle is always $180^{\circ}$. This is a fundamental rule that applies to all triangles regardless of their size or shape.

To visualize this property, imagine cutting out the three corners of a paper triangle (labeled $\angle A$, $\angle B$, and $\angle C$) and arranging them side-by-side at a single point. The three angles will perfectly form a straight line, which represents a straight angle of $180^{\circ}$.

In a right-angled triangle, one angle is always $90^{\circ}$, which is represented visually by a small square at the vertex. This means the sum of the other two acute angles must be $180^{\circ} - 90^{\circ} = 90^{\circ}$. These two angles are called complementary angles.

In an equilateral triangle, all three sides are equal and all three interior angles are equal. Since the total sum must be $180^{\circ}$, each individual angle must measure exactly $\frac{180^{\circ}}{3} = 60^{\circ}$.

In an isosceles triangle, two sides are of equal length, and the angles opposite these sides are also equal. If the measure of the non-equal angle (the vertex angle) is known, the other two angles can be found by subtracting the known angle from $180^{\circ}$ and dividing the result by $2$.

A triangle cannot have more than one right angle or more than one obtuse angle (an angle greater than $90^{\circ}$). If it did, the sum of just those two angles would already meet or exceed $180^{\circ}$, leaving no room for the third angle.

The property can be verified by drawing a line through one vertex of a triangle that is parallel to the opposite side. Using the properties of parallel lines and transversal (alternate interior angles), the three angles at the vertex will lie on a straight line, summing to $180^{\circ}$.