Proving that a Triangle with a Specific Angle is a Right-Angled Triangle

In geometry, one of the fascinating properties of triangles is that the exterior angle of any triangle is equal to the sum of the two opposite interior angles. This property often allows us to solve complex geometric problems by breaking them down into simpler steps. This article explores a special case where a triangle's angle is equal to the sum of the other two angles, proving that the triangle is a right-angled triangle.

Understanding the Triangle Angle Sum Property

The sum of the interior angles of a triangle is always 180°. This fundamental property is crucial for solving numerous geometry problems. When one angle of a triangle is equal to the sum of the other two angles, we can use this sum of angles property to prove that the triangle is a right-angled triangle.

Proof

Consider a triangle ABC with angles A, B, and C. By the triangle angle sum property, we know that:

[A B C 180^{circ}]

We are given that one of the angles, let's say angle A, is equal to the sum of the other two angles:

[A B C]

Substituting the Given Condition into the triangle angle sum equation, we get:

[B C B C 180^{circ}]

Simplifying the equation:

[2B 2C 180^{circ}]

Dividing the entire equation by 2

[B C 90^{circ}]

Conclusion: Since we established that B C 90^{circ}, we can now substitute back into the equation for angle A:

[A B C 90^{circ}]

Thus, angle A is a right angle.

Final Statement

Therefore, if one angle of a triangle is equal to the sum of the other two angles, that angle must be 90^{circ}, proving that the triangle is a right-angled triangle.

Conclusion: This property provides a simple yet powerful tool to identify right-angled triangles in geometric problems. Understanding the triangle angle sum property and its implications can greatly enhance one's problem-solving skills in geometry.

Related Keywords: right-angled triangle, triangle angle sum, angle properties