Finding the Odd Statement About Triangle Properties

The question asks us to identify the statement that is different or "odd" compared to the others, in relation to a triangle. This means three of the statements are generally true for all triangles, while one is either not true for all triangles or is a specific case.

Let's analyze each statement:

Analysis of Statement A: Longest Side and Greatest Angle

Statement A says: "The longest side is opposite to the greatest angle."

• This is a fundamental theorem in geometry. In any triangle, the side opposite the largest interior angle is the longest side, and conversely, the angle opposite the longest side is the largest angle.
• This statement is true for all types of triangles (acute, obtuse, right-angled).

So, Statement A is a general property of triangles.

Analysis of Statement B: Exterior Angle Theorem

Statement B says: "The exterior angle of a triangle = the sum of interior opposite angles."

• An exterior angle of a triangle is formed by extending one side of the triangle.
• The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles (the angles that are not adjacent to the exterior angle).
• This theorem is true for all triangles.

So, Statement B is a general property of triangles.

Analysis of Statement C: Triangle Inequality Theorem

Statement C says: "The sum of 2 sides is greater than the 3rd side."

• This is the Triangle Inequality Theorem. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
• This condition is necessary for a triangle to exist. If this condition is not met, the three segments cannot form a triangle.
• This theorem is true for all triangles.

So, Statement C is a general property of triangles.

Analysis of Statement D: Pythagorean Theorem

Statement D says: "The square of one side = the sum of the squares of the other two sides"

• This statement describes the Pythagorean Theorem, which is expressed as \(a^2 + b^2 = c^2\), where \(c\) is the length of the hypotenuse (the side opposite the right angle), and \(a\) and \(b\) are the lengths of the other two sides (legs).
• This theorem is true only for right-angled triangles. It is not true for all triangles (e.g., an equilateral triangle or an obtuse triangle).

So, Statement D is a property specific to right-angled triangles, not a general property of *all* triangles like statements A, B, and C.

Identifying the Odd Statement

Comparing the statements, A, B, and C are properties that hold true for any triangle. Statement D, on the other hand, is only true for a specific type of triangle (a right-angled triangle).

Therefore, Statement D is the odd statement out because it does not describe a property applicable to all triangles.

Conclusion

Based on the analysis, Statement D is the odd statement. The option corresponding to Statement D is the correct answer.

Revision Table: Key Triangle Properties

Additional Information: Types of Triangles

Triangles can be classified based on their sides or angles:

Classification by Sides:

• Equilateral Triangle: All three sides are equal in length. All three angles are also equal (\(60^\circ\)).
• Isosceles Triangle: At least two sides are equal in length. The angles opposite the equal sides are also equal.
• Scalene Triangle: All three sides have different lengths. All three angles have different measures.

Classification by Angles:

• Acute Triangle: All three angles are acute angles (less than \(90^\circ\)).
• Right-Angled Triangle: Has one right angle (\(90^\circ\)). The side opposite the right angle is called the hypotenuse.
• Obtuse Triangle: Has one obtuse angle (greater than \(90^\circ\)).

Understanding these different types helps clarify why certain properties, like the Pythagorean theorem, apply only to a specific type.