The sum of the interior angles of a triangle is a fundamental property in Euclidean geometry.
Here's a detailed explanation of why this is true:
1. Basic Definition of a Triangle:
- A triangle is a polygon with three edges and three vertices.
2. Sum of Interior Angles in a Polygon:
- For any polygon with [tex]\( n \)[/tex] sides, the sum of the interior angles can be calculated using the formula:
[tex]\[
\text{Sum of interior angles} = (n-2) \times 180^\circ
\][/tex]
- For a triangle, [tex]\( n = 3 \)[/tex].
3. Applying the Formula to a Triangle:
- Substitute [tex]\( n = 3 \)[/tex] into the formula:
[tex]\[
\text{Sum of interior angles} = (3-2) \times 180^\circ = 1 \times 180^\circ = 180^\circ
\][/tex]
- Hence, the sum of the interior angles of a triangle is 180 degrees.
4. Fundamental Theorem:
- This result is consistent with the Triangle Sum Theorem, which states that the interior angles of a triangle add up to [tex]\( 180^\circ \)[/tex].
5. Practical Examples:
- Consider a right-angled triangle with angles [tex]\( 90^\circ, 45^\circ, \)[/tex] and [tex]\( 45^\circ \)[/tex]. Summing these angles gives:
[tex]\[
90^\circ + 45^\circ + 45^\circ = 180^\circ
\][/tex]
- For an equilateral triangle, each angle is [tex]\( 60^\circ \)[/tex]. Summing these gives:
[tex]\[
60^\circ + 60^\circ + 60^\circ = 180^\circ
\][/tex]
Based on the above points, the statement "The sum of the interior angles of a triangle is 180°" is indeed True.
