A triangle is a polygon with three sides and three angles. It is one of the foundational shapes in geometry. The sum of the interior angles of any triangle is always equal to \(180^\circ\).
Types of Triangles by Side Length
- Equilateral Triangle: All three sides are equal in length, and all three internal angles are equal to \(60^\circ\).
- Isosceles Triangle: Exactly two sides are equal in length, and the angles opposite those sides are also equal.
- Scalene Triangle: All three sides have different lengths, and all three internal angles have different measures.
Types of Triangles by Angle
- Acute Triangle: All three interior angles are less than \(90^\circ\).
- Right Triangle: Exactly one interior angle is a right angle (\(90^\circ\)). The side opposite the right angle is the hypotenuse.
- Obtuse Triangle: Exactly one interior angle is greater than \(90^\circ\) (obtuse).
Basic Properties
- The sum of the interior angles is always \(180^\circ\).
- Triangle Inequality Theorem: The length of any one side of a triangle is always less than the sum of the lengths of the other two sides.
- Every triangle possesses a perimeter (calculated as the sum of all side lengths) and an enclosed surface area.
Examples
Example 1: Find the missing interior angle
Problem: A triangle has interior angles measuring \(50^\circ\), \(60^\circ\), and one unknown angle. What is the missing angle?
Solution:
Since the sum of interior angles is \(180^\circ\):
\[
180^\circ - (50^\circ + 60^\circ) = 180^\circ - 110^\circ = 70^\circ
\]
The missing angle is \(70^\circ\).
Example 2: Check triangle inequality for given sides
Problem: Can a triangle have side lengths of 5 cm, 7 cm, and 13 cm?
Solution:
Check the triangle inequality by adding the two shorter sides:
\[
5 + 7 = 12
\]
Since 12 is less than 13, the sum of the two shorter sides does not exceed the longest side.
Answer: No, such a triangle is not possible.
Example 3: Calculate the area of a triangle
Problem: Find the area of a triangle with a base of 8 cm and a corresponding height of 5 cm.
Solution:
Use the standard triangle area formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substitute the given values:
\[
A = \frac{1}{2} \times 8 \times 5 = 20 \, \text{cm}^2
\]
The area is \(20 \, \text{cm}^2\).