Welcome! Today we'll explore one of the most fundamental properties in geometry: why the interior angles of any triangle always add up to 180 degrees. We'll use a triangle ABC to demonstrate this beautiful geometric proof using parallel lines and alternate interior angles.
Now we begin our proof. The first step is to draw a line through vertex A that is parallel to the base BC. This parallel line is crucial because it will help us establish relationships between the angles using the properties of parallel lines and transversals.
Now we identify the alternate interior angles. When the parallel line through A is crossed by the sides AB and AC acting as transversals, we create alternate interior angles. Angle 1 equals angle B, and angle 2 equals angle C. This is a fundamental property of parallel lines.
Now comes the crucial step. The three angles on the straight line through vertex A must sum to 180 degrees, because they form a straight angle. Since we proved that angle 1 equals angle B and angle 2 equals angle C, we can substitute to get: angle B plus angle A plus angle C equals 180 degrees.
To summarize what we've learned: The interior angles of any triangle always add up to 180 degrees. This fundamental property is proven using parallel lines and alternate interior angles. This geometric principle applies universally to all triangles and forms the foundation for many other geometric concepts.