What Are the Angles of a Triangle?

Angle of a triangle (or any other polygon) is defined as the angle between two lines joining points A and B.

The following diagram shows the three sides of a triangle.

In this figure, the line from point A to point B is called the “horizontal” line; and the line from point B to point C is called the “vertical” line. If we draw these lines parallel to each other, they form a right angled triangle with one side being longer than another.

If we take the horizontal line and extend it outwards, we will get a straight line which represents the vertical line. Now if we connect all the points along this straight line, then we will have formed a triangle. This means that there are four different angles that can be represented by this type of shape. These four angles are known as the “angles of a triangle”.

Let’s say that the horizontal line is drawn at 90 degrees from the vertical line. Then the angle between them would be 45°.

If we take a closer look at this diagram, it will become clear that if we draw a straight line from point A to point B, then its angle with respect to the vertical line would be 90°.

The right angle is the angle between the horizontal and vertical lines. This is represented by letter “A” in the above diagram. The other three angles are known as the acute angles, which can also be represented by the letters “B”, “C” and “D”. The right angle (A) always measures 90 degrees and each of the acute angles are lesser than 90 degrees.

The sum of all four angles of any triangle is equal to 270 degrees.

This means that the angle between the vertical and horizontal lines would always be 90°. The full angle (measuring from 0° to 360°) of this right angled triangle represents the sum of angles of a triangle.

We can also say that the sum of all angles in a triangle is equal to 180 degrees.

The following diagram shows a right angled triangle with its three angles identified.

When two lines are drawn intersecting each other, then the four angles formed are called the “angles of a quadrilateral”. If all these angles are equal (i.e. all of them are 90°), then it is known as a “square”.

On the other hand, if all the angles are unequal (i.e. none of the angles are equal) then it is known as a “oblong”.

In the above diagram, the angle marked as (A) is the right angle and (B), (C) and (D) are the acute angles. If we take a line from each of these acute angles up to the horizontal line, then we will form three equal sized isosceles triangles. Since the sum of all the angles of any triangle is 180 degrees, then all the angles of an isosceles triangle must also add up to 180 degrees.

The following diagram shows a “square” and an “oblong”.

In the above diagrams, it can be seen that the side lengths of both the figures are in the ratio of 1:2:2:1. The square has all its sides equal and all its angles equal. The oblong has unequal sides and unequal angles.

The following diagram shows an isosceles triangle with its angles identified.

All the above facts about the right angled triangles and isosceles triangles are important because we will be using them a lot in our future lessons.

In the next lesson, we will look at some simple formulas for finding out information about the angles of a triangle.

__Sources & references used in this article:__

- Triangles with integer dimensions (NJ Gilbertson, KC Rogers – The Mathematics Teacher, 2016 – pubs.nctm.org)

https://pubs.nctm.org/downloadpdf/journals/mt/109/9/article-p654.pdf - Pythagoras Triples Explained via Central Squares. (LT Gomes – Australian Senior Mathematics Journal, 2015 – ERIC)

https://eric.ed.gov/?id=EJ1093370 - Three Old Babylonian methods for dealing with” Pythagorean” triangles (E Robson – Journal of Cuneiform Studies, 1997 – journals.uchicago.edu)

https://www.journals.uchicago.edu/doi/pdfplus/10.2307/1359891