In This Triangle Which Of The Following Is True

In This Triangle: Which of the Following is True? A Deep Dive into Triangle Properties

This article explores the properties of triangles, focusing on how to determine which statements are true given specific information about a triangle's angles and sides. Understanding triangle properties is fundamental in geometry and crucial for solving various mathematical problems. This guide will cover key concepts and provide examples to help you master this topic. We'll cover isosceles, equilateral, scalene, and right-angled triangles, and how to identify their defining characteristics.

Understanding Basic Triangle Properties

Before we delve into specific problems, let's refresh our understanding of fundamental triangle properties:

• Angle Sum Property: The sum of the interior angles of any triangle always equals 180 degrees. This is a cornerstone principle in triangle geometry.
• Side-Angle Relationship: The longest side of a triangle is always opposite the largest angle, and vice-versa. The shortest side is opposite the smallest angle.
• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem prevents the construction of impossible triangles.

Types of Triangles and Their Properties

Triangles are classified based on their sides and angles:

• Equilateral Triangles: All three sides are equal in length, and all three angles are equal (60 degrees each).
• Isosceles Triangles: Two sides are equal in length, and the angles opposite these sides are also equal.
• Scalene Triangles: All three sides are of different lengths, and all three angles are of different measures.
• Right-Angled Triangles: One angle measures 90 degrees. The side opposite the right angle is called the hypotenuse, and it's always the longest side. The Pythagorean theorem applies to right-angled triangles (a² + b² = c², where a and b are the legs and c is the hypotenuse).

Determining the Truth Value of Statements about Triangles

Let's tackle how to determine whether a statement about a triangle is true or false. This often involves applying the properties discussed above. Consider the following examples:

Example 1:

A triangle has angles measuring 40°, 60°, and 80°. Which of the following is true?

• a) It is an isosceles triangle.
• b) It is a right-angled triangle.
• c) It is an equilateral triangle.

Solution:

The sum of the angles (40° + 60° + 80° = 180°), satisfying the angle sum property. Since all angles are different, it's a scalene triangle. Therefore, only statement (a) is incorrect; it's not an isosceles triangle. Statements (b) and (c) are also false.

Example 2:

A triangle has sides of length 5cm, 12cm, and 13cm. Which of the following is true?

• a) It is a right-angled triangle.
• b) It is an acute-angled triangle.
• c) It is an obtuse-angled triangle.

Solution:

Let's check if the Pythagorean theorem holds: 5² + 12² = 25 + 144 = 169 = 13². Since a² + b² = c², this is a right-angled triangle. Therefore, statement (a) is true. Statements (b) and (c) are false.

Conclusion:

Determining the truth of statements about triangles relies on a solid understanding of their properties. By applying the angle sum property, side-angle relationships, the triangle inequality theorem, and the definitions of different triangle types, you can accurately analyze and solve problems related to triangle characteristics. Remember to systematically check each statement against the given information and the relevant theorems. Practice with various examples will solidify your understanding and improve your problem-solving skills.