Triangles And Their Parts

A triangle is a polygon with three sides, three vertices and three angles.


Altitude: It's easy to find the height of a rectangle or a square, but what about a triangle? The Altitude or height is calculated by dropping a perpendicular from a vertex to its opposite side. The dotted blue lines the above triangle are, the perpendicular distance between vertex and the opposite side, called altitude or height of the triangle \large ABC.

Interior Angles: As the name suggests these are the angle which lie inside the triangle. As mentioned earlier there are a total 3 of them for any triangle. You can see the interior angles marked as dotted red line in the above shown figure.

Exterior Angles: When a side is extended to a point outside the triangle, the angle formed between the extended side and lateral side is called an exterior angle to the corresponding angle.

Median: A median is a line segment joining a vertex and the middle point of its opposite side. There are three sides in a triangle, for every side we have a middle point. For a side \large XY in a triangle \large XYZ, the midpoint of \large XY is joint to the opposite vertex \large Z, the line segment formed is a median. 


Classification Of Triangles

I. Triangles can be classified into 3 types on the basis of their side lengths, these are

Equilateral Triangle: Triangle having all three sides equal as well as all the three angles. Each angle of an equilateral triangle is \large 60^o.

Isosceles Triangle: Triangle having two sides equal and the opposite angles, with respect to the equal sides, equal. 

Scalene Triangle: Triangle having all three sides different  and all three angles different are called Scalene triangles.

Image showing equilateral, isosceles and scalene triangle_makeageek


II. Triangles can also be classified into 3 types on the basis of their internal angle measurements, these are

Acute Angled Triangle: Triangle with all three angles acute (less than \large 90^o)

Right Angled Triangle; Triangle with one of the angles as a right angle (equal to \large 90^o)

Obtuse Angled Triangle: Triangle with one obtuse angle (greater than \large 90^o)



Sum of Angles Of A Triangle

Figure showing interior and exterior angles of a triangle and mathematical relationships between these.


I. The sum of all interior angles of a triangle, \large \angle a + \angle b + \angle c = 180 \degree.


II. The sum of the exterior angles of a triangle\large \angle X + \angle Y + \angle Z = 360 \degree.

Isosceles Triangle

In an isosceles triangle, the angles opposite the equal sides have the same measure. If two sides of a triangle are equal, then the angles opposite those sides are equal. Similarly, if two angles of a triangle are equal, then the sides opposite those angles are equal. 




Right Angled Triangle and Pythagorean Theorem

I. In a right triangle, the side of the triangle opposite the right angle is called the hypotenuse and the other two sides are called the legs of the triangle.





The Pythagorean Theorem  tells us that in any right angled triangle, the sum of the squares of the legs equals the square of the hypotenuse. In mathematical terms, \LARGE a^2+b^2=c^2





If we multiply all three sides lengths of a right triangle by the same positive number, then the three new side lengths also satisfy the Pythagorean Theorem. In other words, if side lengths a, b and c satisfy \large a^2+b^2=c^2 , then \large (na)^2+(nb)^2=(nc)^2 for  any positive integer n. 


II. In an isosceles right triangle,  the legs are congruent (equal) and the hypotenuse is \large \sqrt{2} times as long as each leg. An isosceles triangle is often called a \large 45-45-90 triangle. 


III. In a right triangle, with acute angles of \large 30^o and \large 60^o, the side lengths are in the ratio \large 1 : \sqrt{3} : 2. Such a triangle is often called a \large 30-60-90 triangle.