A triangle is a basic geometric form with three sides and three corners. Each side links to two adjacent sides, resulting in three corners where the sides meet. The angles within a triangle always sum to 180 degrees. Triangles are classified into three types: equilateral (all sides and angles are equal), isosceles (two sides and two angles are equal), and scalene (all sides and angles differ).
Consider a piece of pizza that has three sides and three corners, similar to a triangle. Another example is a street sign with three edges that intersect at each corner. Triangles are fundamental forms seen in many common items, and they are required in geometry to comprehend the basic concepts of angles and measures.
Some of the important properties of triangles are added below:
The properties of triangles in geometry are:
- Angle sum property.
- Triangle inequality property.
- Pythagoras theorem.
- Side-angle relationship.
- Exterior angles property.
- Congruence conditions.
Various properties of triangles are discussed in detail below:
Angle Sum Property
Angle Sum Property is a fundamental property in geometry that asserts that the sum of all angles within a triangle is always 180 degrees. This technique is useful for solving for missing angles or determining triangle validity. For example, if two angles are 60 degrees each, the third angle must also be 60 degrees to meet this criterion.
Angle 1 + Angle 2 + Angle 3 = 180∘
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Triangle Inequality Property
The total of any two sides of a triangle exceeds the length of the third side. In other words, the shortest path between two places is a straight line. This is expressed as:
a + b > c
Where a, b, and c are the lengths of the sides of the triangle.
Pythagoras Property
In a right triangle, the square of the hypotenuse's length (the side opposite the right angle) equals the sum of the squares of the other two sides. This is called the Pythagorean theorem.
c2 = a2 + b2
Hypotenuse length is denoted by c, whereas the other two sides' lengths are denoted by a and b.
Side Opposite the Greater Angle is the Longest Side
The side opposite the greatest angle in a triangle is the longest side. This is an observable property, not a formal theorem. When given a triangle's angles, it helps to determine which side is the longest.
Exterior Angle Property
Each exterior angle of a triangle equals the sum of its two remote interior angles. The mathematical expression is:
Exterior Angle = Sum of Remote Interior Angles
Congruence Property
Triangles of the same size and shape are said to be congruent. This attribute is useful in assessing if two triangles are identical. It may be proved by congruence criteria such as
- Side-Side-Side (SSS)
- Side-Angle-Side (SAS)
- Angle-Side-Angle (ASA)
- Right Angle-Hypotenuse-Side(RHS)
These qualities are essential for understanding and solving triangle-related geometry issues.
Here are the formulae for triangles presented in a table format:
These formulae are commonly used in geometry to compute the characteristics of triangles.
Examples of Properties of Triangle
Example 1: The sides of a triangle are 6 cm, 7 cm, and 9 cm. Find the perimeter and semi-perimeter of the triangle.
Solution:
Sides of triangle are a = 6 cm, b = 7 cm, and c = 9 cm.
To calculate the perimeter of a triangle, use the formula P = a + b + c,
P = 6 + 7 + 9 = 22 cm
Therefore, the perimeter of the supplied triangle is 22 cm.
To calculate the semi perimeter of a triangle, use the formula P = a( + b + c) / 2,
P = (6 + 7 + 9) / 2
P = 22/2 = 11 cm
Therefore, the semi perimeter of the supplied triangle is 11 cm.
Example 2: The measure of two angles in a triangle is 75∘ and 85∘. What will be the third angle's measurement?
Solution:
Angles of a triangle are measured as 75∘ and 85∘.
Sum of two angle measurements 75∘ + 85∘ = 160∘
Total of all three angles in a triangle = sum property equals 180 degrees.
Hence, the measure of the third angle = 180∘−160∘=20∘.
Example 3: A triangle with sides of 6 cm, 8 cm, and 9 cm (with 8 cm as the base) has an altitude of 5.5 cm. Calculate the area of the triangle.
Solution:
Given:
- Base = 8 cm
- Height = 5.5 cm
Area of a Triangle(A) = 1/2 × b(base) × h(height)
A = (1/2) × 8 × 5.5
A = 22 cm2
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