A triangle is a polygon with three sides and three interior angles.
One of the most important properties in geometry is that the sum of the interior angles of any triangle is 180 degrees.
This rule helps us find missing angles and solve many geometry problems.
In this lesson, we will prove why the sum of the interior angles of a triangle equals 180°.
Key Ideas Used in the Proof
To prove this property, we use two basic geometry facts:
- A straight angle measures 180°.
- Alternate interior angles are equal when two parallel lines are cut by a transversal.
Proof Idea
Consider a triangle with interior angles:
a, b, and c.
Extend one side of the triangle to form a straight line.
Then draw a line through the opposite vertex that is parallel to the base of the triangle.
Using the properties of alternate interior angles, we find that the angles formed on the straight line correspond to the triangle’s angles.
Since a straight line measures 180°, the three interior angles of the triangle must satisfy:
a + b + c = 180°
Therefore, the sum of the interior angles of a triangle is 180 degrees.
Example
Two angles of a triangle are 50° and 60°.
Find the third angle.
50° + 60° + x = 180°
110° + x = 180°
x = 70°
So the third angle is 70°.
Video Explanation
Practice
Continue Learning
- Types of Triangles
- Triangles (Basics)
- Sum of Angles of a Triangle is 180° (Proof)
- Find the Missing Angle of a Triangle
- Sum of Exterior Angles of a Triangle (Proof)
- Exterior Angle Theorem of a Triangle
- Two Exterior Angles at a Vertex are Equal (Proof)
- Two Exterior Angles are Equal at a Vertex (Quick Proof)