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Classification and Properties of Triangles

A triangle is a geometrical figure that has three sides and three angles. Do you remember a polygon with the least number of sides? Well, that is a triangle. A polygon is a simple closed curve. It is made of line segments. Can we make a closed curve with a single line segment? The answer is no. Even if we try to make a curve with two line segments, it won’t be possible. So we need to make three segments and intersect them to form a closed curve, that is, a triangle. So, a triangle is a closed curve made up of three line segments. 

A square comprises four lines of the same length and four right angles. Constructing a triangle is easier because there are no rules. If we join three line segments in whichever way, we get a triangle.

Let us see the different types of triangles we can get.

Activity

This activity will help us understand how to classify triangles on a fundamental level. 

  • We take images of different triangles from the internet.
  • Then we print them.
  • We find the measurement of different sides and angles of the given triangles using a protractor and a ruler.
  • Will all the sides and angles of different triangles be equal? The answer is no.
  • The sides and angles of different triangles vary in measurement.
  • This difference in the measurement of the sides and measurements of different triangles helps us classify them.
  • We also notice that if all the angles of a triangle are equal, then its sides are also equal. 
  • If all the sides in a triangle are equal, its sides are equal.
  •  If two sides of a triangle are equal, it has two equal angles, and if two angles of a triangle are equal, then it has two equal sides. 
  • If none of the triangle angles is equal, none of the sides is equal.
  • If the three sides of a triangle are unequal, the three angles are also unequal.
  • After going through the whole discussion, we will be able to classify each triangle by its angles and sides.

Properties of triangles

  1. A triangle has three sides, three angles, and three vertices.
  2. The sum of the interior angles of a triangle is equal to 180 degrees.
  3. The sum of two sides of a triangle must be greater than the third side.
  4. The side of the triangle, contrary to the biggest angle, is the largest.
  5. If we add the values of the three exterior angles of a triangle, the result is 360 degrees.

Types of angles based on their measurement

We need to understand the types of angles before classifying the triangles by angles. It will help us understand the discussion on how to classify triangles better.

  • An angle that is less than 90 degrees is called an acute angle.
  • An angle whose measurement is equal to 90 degrees is called a right angle.
  • An angle whose measurement is greater than 90 degrees and lesser than 180 degrees is called an obtuse angle

How to classify triangles based on their angles

A triangle has three angles. If we add up all three angles of a triangle, the result must be 180 degrees.

Like we divided the types of angles on the same basis, we can divide the triangle into different types:

  • Acute-angled triangle: A triangle with all three acute angles is called an acute-angled triangle.
  • Right-angled triangle: A triangle with one right angle is called a right-angled triangle. A triangle can have only one right angle. The sum of all three angles of a triangle must be 180 degrees. So, if one angle is equal to 90 degrees, the sum of the other two angles has to be 90 degrees. If we try making a triangle with two right angles, we can’t assume the third angle equals zero. In that case, the third side of the triangle will overlap the other two sides, and a triangle can not be formed. So, it is impossible to construct a triangle with more than one angle equal to 90 degrees. The other two angles of a right-angled triangle have to be acute.
  • Obtuse angled triangle: In such a triangle, the measurement of one of the angles is more than 90 degrees but less than 180 degrees. Can we now say that an obtuse-angled triangle is one whose one of the angles is an obtuse angle? Yes, we can. As the sum of all three angles of a triangle has to be equal to 180 degrees, the other two angles of an obtuse triangle have to be acute.

How to classify triangles based on their sides

A triangle is made up of three sides. The three sides of the triangle may have the same measurement, or their measurement may be different.

  • Isosceles triangle: An isosceles triangle is one whose two sides have the exact measurement, but the third side has a different measurement.
  • Scalene triangle: A scalene triangle is one whose all three sides have different measurements.
  • Equilateral triangle: An equilateral triangle is one whose all three sides have the same measurement. The equal sides of a triangle are called congruent sides. So, all the sides of an equilateral triangle are congruent.

Key Table – Classify the following triangle and check all that apply 

Exterior angles of a triangle: 

When a triangle’s side is extended, it forms an angle that lies outside the triangle. This angle is called the exterior angle of the triangle. The Exterior Angle Theorem gives this concept. Let us construct a triangle with its exterior angle to understand this concept better.

  • We draw a triangle ABC and extend one of its sides, let us say BC as shown in the figure.
  • We observe the angle ACD formed at point C. This angle lies in the exterior of the triangle ABC.
  • We can call it an exterior angle of the triangle ABC formed at vertex C.
  • Angle BCA is an adjacent angle to angle ACD.
  • The remaining two angles of the triangle, angle A and angle B, are called the two interior opposite angles or the two remote interior angles of angle ACD.
  • A triangle’s exterior angle equals the combined value of its opposite interior angles.

Applications of triangles in our lives 

Triangle is the basic unit of all polygons. The concept of triangles has been used extensively since ancient times. Triangles appear in many aspects of our everyday lives like engineering, Mathematics, architecture, carpentry, astronomy, navigation, and Physics. This shape is seen almost everywhere. A triangle is the strongest shape forming a strong base. 

  1. In architecture: Spotting triangular structures in the making of buildings is not an unusual sight. The application of the concept of triangles in making buildings can be seen in the form of the Pyramids in Egypt. Another such example is the historical Eiffel Tower. The triangular shape makes these historical buildings look unique and attractive. Also, it provides a strong foundation for the base of the tower. A triangle derives its strength from its shape, which spreads equal forces between its three sides. We also see so many structures having a square shape. That is because a square is easily stackable. Building triangular buildings is trickier, but the resulting structure is more stable.
  2. The roofs of most houses are triangular. Houses with triangular roofs are more commonly seen in areas where it snows. The triangular roofs create a slope that allows the snow to fall and prevents water stagnation on the roofs.
  3. The sail of a sailing boat has a triangular shape. Earlier sailing boat sails used to be square, but now almost all are triangular. Their shape helps the boat travel against the direction of the wind. This method is called tacking. 
  4. Mountains are triangular. We use the concept of the properties of a right triangle to find the height of a hill or a pole. We can also find out the ship’s distance from a tower using the concept of triangles.
  5. Triangular shapes are used in the building of bridges. This shape helps to distribute the weight without interfering with the proportions evenly. Incorporating triangular shapes in the structure of bridges has helped them become strong and endure more weight.
  6. The knowledge of right angles is employed in the construction of staircases. The ladder also makes a triangle when placed against the wall. This position holds it steadily.
  7. Many food items like sandwiches, pizzas, and packed snacks are triangular. This shape makes them look more inviting and easy to handle. Children prefer eating triangular sandwiches compared to sandwiches of other shapes. 
  8. Similar triangles are used to determine the height of objects that are difficult to measure manually. For example, tall buildings, towers, etc.
  9. A slope forms a triangle with the ground. It makes loading and unloading work easier. We have all noticed how the luggage is unloaded from an airplane on a slope. It makes the job easier.
  10. Traffic signboards are triangular. It makes them easily identifiable.
  11. Our geometrical instruments, like protractors, are triangular. They help us draw parallel lines, vertical lines, and other angled lines accurately.

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