Triangle Angle Relationships

Triangles are fundamental geometric shapes formed by three straight sides and three angles. Understanding the relationships between these angles and sides is essential for solving complex geometry problems.

The Interior Angle Sum Theorem

The most important rule to remember about triangles is the Interior Angle Sum Theorem. It states that the sum of the three interior angles of any triangle is always exactly $180^\circ$.

If a triangle has angles $A$, $B$, and $C$, the formula is: $\angle A + \angle B + \angle C = 180^\circ$

Example: In $\triangle ABC$, you are given $\angle A = 50^\circ$ and $\angle B = 70^\circ$. Find $\angle C$.

To find $\angle C$, subtract the known angles from $180^\circ$: $\angle C = 180^\circ - (50^\circ + 70^\circ)$ $\angle C = 180^\circ - 120^\circ = 60^\circ$

The Exterior Angle Theorem

An exterior angle is formed when you extend one side of a triangle outward. The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent (opposite) interior angles.

Example: Using the same $\triangle ABC$ where $\angle A = 50^\circ$ and $\angle B = 70^\circ$, find the exterior angle at $C$.

The two non-adjacent interior angles to $C$ are $A$ and $B$. $\text{Exterior } \angle C = \angle A + \angle B$ $\text{Exterior } \angle C = 50^\circ + 70^\circ = 120^\circ$ (Notice that the interior $\angle C$ ($60^\circ$) and the exterior $\angle C$ ($120^\circ$) form a straight line, adding up to $180^\circ$.)

Classifying Triangles

Triangles can be classified in two ways: by their angles and by their side lengths.

Classification by Angles

• Acute Triangle: All three interior angles are less than $90^\circ$.
• Right Triangle: Exactly one angle is equal to $90^\circ$ (a right angle).
• Obtuse Triangle: Exactly one angle is greater than $90^\circ$.

Classification by Sides

• Equilateral Triangle: All three sides are equal in length. As a result, all three angles are also equal (each is exactly $60^\circ$).
• Isosceles Triangle: Exactly two sides are equal in length. The angles opposite these equal sides (called base angles) are also equal.
• Scalene Triangle: All three sides have different lengths, meaning all three angles are also different.

Example: Classify a triangle with side lengths of $5$, $5$, and $8$.

Because exactly two sides are equal ($5$ and $5$), this is an isosceles triangle.