Euclidean Geometry: Triangles

Introduction

Abongile Mkonjeni 207113131

Definition of Triangle

a triangle is a three-sided polygon that consists of three edges and three vertices. The most important property of a triangle is that the sum of the internal angles of a triangle is equal to 180 degrees. This property is called angle sum property of triangle.

https://cdn1.byjus.com/wp-content/uploads/2020/01/Definition-of-Triangle.png

 

If ABC is a triangle, then it is denoted as ΔABC, where A, B and C are the vertices of the triangle. A triangle is a two-dimensional shape, in Euclidean geometry, which is seen as three non-collinear points in a unique plane.

 

 

Angles of Triangle
There are three angles in a triangle. These angles are formed by two sides of the triangle, which meets at a common point, known as the vertex. The sum of all three interior angles is equal to 180 degrees.
If we extend the side length outwards, then it forms an exterior angle. The sum of consecutive interior and exterior angles of a triangle is supplementary

https://cdn1.byjus.com/wp-content/uploads/2020/07/Exterior-angles.png

 

Types of Triangles
Triangles can be broadly classified into two types, which are:
• Triangles based on the lengths of their sides
• Triangles based on their interior angles
These two triangle types are explained here along with their further classifications.

Based on their Sides Based on their Angles
Scalene Triangle Acute Triangle
Isosceles Triangle Obtuse Triangle
Equilateral Triangle Right Triangle

Types of Triangles Based on Sides
According to the lengths of their sides, triangles can be classified into three types which are:
• Scalene
• Isosceles
• Equilateral

Scalene Triangle

A scalene triangle has all side lengths of different measures. No side will be equal in length to any of the other sides in such a triangle. In a scalene triangle, all the interior angles are also different. The figure given below illustrates a scalene triangle. You can see how none of the sides is equal in length.

https://cdn1.byjus.com/wp-content/uploads/2020/10/Triangles-01.png

Isosceles Triangle
In an isosceles triangle, the lengths of two of the three sides are equal. So, the angles opposite the equal sides are equal to each other. In other words, an isosceles triangle has two equal sides and two equal angles. The figure given below illustrates an isosceles triangle.

https://cdn1.byjus.com/wp-content/uploads/2020/10/Triangles-02.png

Equilateral Triangle
In an equilateral triangle, all the lengths of the sides are equal. In such a case, each of the interior angles will have a measure of 60 degrees. Since the angles of an equilateral triangle are same, it is also known as an equiangular triangle. The figure given below illustrates an equilateral triangle.

https://cdn1.byjus.com/wp-content/uploads/2020/10/Triangles-03.png

Types of Triangles Based on Angles
Triangles can be classified into three types with respect to their interior angles which are:
• Acute-angled
• Obtuse-angled
• Right-angled

Acute Triangle
An acute triangle is a triangle who’s all the three interior angles are acute. In other words, if all interior angles are less than 90 degrees, then it is an acute-angled triangle. The figure given below illustrates an acute triangle.

https://cdn1.byjus.com/wp-content/uploads/2020/10/Triangles-04.png

Obtuse Triangle
Obtuse triangles are those in which one of the three interior angles has a measure greater than 90 degrees. In other words, if one of the angles in a triangle is an obtuse angle, then the triangle is called an obtuse-angled triangle. The figure given below illustrates an obtuse triangle.

https://cdn1.byjus.com/wp-content/uploads/2020/10/Triangles-06.png

Right Triangle
A right triangle is a triangle in which one of the angles is 90 degrees. In a right-angled triangle, the side opposite to the right angle (90-degree angle) will be the longest side and is called the hypotenuse. You may come across triangle types with combined names like right isosceles triangle and such, but this only implies that the triangle has two equal sides with one of the interior angles being 90 degrees. The figure given below illustrates a right triangle.

https://cdn1.byjus.com/wp-content/uploads/2020/10/Triangles-05.png

Area of a Triangle
The area of a triangle is the region occupied by the triangle in 2d space. The area for different triangles varies from each other depending on their dimensions. We can calculate the area if we know the base length and the height of a triangle. It is measured in square units.
Suppose a triangle with base ‘B’ and height ‘H’ is given to us, then, the area of a triangle is given by-

https://cdn1.byjus.com/wp-content/uploads/2020/10/Triangles-07.png 

Area of triangle = Half of Product of Base and Height

Area = 1/2 × Base × Height

use the link below for more details on triangle and how to classify triangles.

https://www.youtube.com/watch?v=mLeNaZcy-hE&t=1s

 

Task

Click the link below and to watch a video that will assist you in completing the task below on Triangles.

(26) Exterior Angle Theorem For Triangles, Practice Problems - Geometry - YouTube

also use the following website for more information.

https://www.siyavula.com/read/maths/grade-12/euclidean-geometry/08-euclidean-geometry-04

5.1 Triangles, quadrilaterals, circles and others | Geometry of shapes | Siyavula

 

Questions

  1. Suleka is walking in east direction. After walking for 55m, she takes a right turn, and walks for another 48m. She is at a distance of.........meters from starting point.
  2. A triangle has three angles, which we call the first, second and third angle here. The second angle is twice the first angle. The third angle is 42°. The first angle is.
  3. If 2 angles in a triangle add up to 124° then what is the value of the third angle?
  4. Harsh is walking in north direction. After walking for 5m, he takes a right turn and walks for another 9m. He then takes a left turn and walks for 7m. He again takes a right turn and walks for another 7m. How far is he form starting point?
  5. In a triangle if each angle is less than sum of other two angles. What is the type of this triangle?
Process
  1. Read the introduction to familiarize yourself with the types of triangle and their properties.
  2. Watch all the videos to be able to answer the questions above.
  3. all answer to be rounded off to 2decimal places.

 

(26) solving problems using triangle properties - YouTube

Evaluation
Questions Angle mark Sides mark Total
1 2 5 7
2 7 0 7
3 5 0 5
4 0 9 9
5 3 0 2
      Total = 30marks 

 

Conclusion

in this lesson we have covered:

  • The types of triangles and their properties.
  • we applied the properties of triangles to solve problems on triangles.
  • On our next lesson we continue with triangles doing the Area rule; Sine rule and Cosine rule.

Watch the video below to prepare for the next lesson.

https://youtu.be/l8PrdgrIA1Y

Credits

Use the link below to learn more on triangles.

(26) How To Calculate The Missing Angle In a Triangle - YouTube

Teacher Page

To solve problems on triangles you need to understand the properties of the different types of Triangle. To find the interior and the exterior angles of triangles and the side 

Abongile Mkonjeni

207113131