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Prove that the sum of all the interior angles of any triangle is 180° using paper cutting and pasting method

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Geometry is a branch of mathematics that is used to study the forms, angles, measurements, proportions of ordinary objects. There are two types of forms available in geometry that is a two-dimensional form which includes flat shapes like rectangle, square, circle, etc., and a three-dimensional form which includes 3D shapes like cuboid, cone, etc. Using geometric principles we can find the field, volume, circumference of the shapes. 

Triangle

Triangle is a closed figure which is formed by three line segments. It consists of three angles and three vertices. The angles of triangles can be the same or different depending on the type of triangle. There are different types of triangles based on lines and angles properties.

Properties of a Triangle

  • Each triangle has 3 sides and 3 angles.
  • The sum of all the angles of a triangle is 180°
  • The perimeter of a triangle is the sum of all three sides of the triangle.
  • A triangle has 3 vertices.

Types of triangles based on angle properties

1. Acute Angled Triangle: Acute angled triangle is a type of triangle in which all the angles of the triangle are less than 90 degrees. All the sides of an acute-angled triangle can be of the same or different lengths.

2. Obtuse Angled Triangle: Obtuse angled triangle is a type of triangle in which one of the angles of the triangle is greater than 90 degrees. All the sides of the obtuse-angled triangle are of different lengths.

3: Right Angled Triangle: Right Angled Triangle is a type of triangle in which one of the angles of a triangle is equal to 90 degrees. This triangle follows the Pythagoras theorem.

How can we prove that the sum of the measures of the angles of any triangle is 180 degrees using paper cutting and pasting?

Solution:

Step 1: Take any triangle (acute, obtuse, right angle), let us take an acute-angled triangle.

Step 2: Mark the angles as x°, y°, and z° respectively.

Step 3: Cut all three angles of the triangle.

Step 4:  Now draw a straight line on a sheet.

Step 5: Arrange the angles which have been cut from the triangle in a manner to form a semi-circle.

Step 6: As we know that measure of a semi-circle is 180 degrees and hence the sum of angles of a triangle is 180 degrees.

x° + y° + z° = 180°

Sample Questions

Question 1: Two angles of a triangle are 108° and 32° find the third angle.

Solution:

As we already know that sum of all the angles of a triangle is 180°

so consider x = 108° and y = 32° and we will find z

By using angle sum formula 

 x° + y° + z° = 180°

108° + 32° + z = 180°

z + 140° = 180°

z = 180° – 140°

z = 40°

Question 2: One angle of an isosceles triangle is 40° and the other is 100°, find the third angle.

Solution:

We know that two angles in an isosceles triangle are equal so either 40° or 100° is the third angle but:

If we take 100° as the third angle then the sum of all three angles will be greater than 180°

Hence the third angle is 40°

Question 3: Two angles of a triangle are 60° and 30° find the third angle.

Solution:

As we already know that sum of all the angles of a triangle is 180°

so consider x = 60° and y = 30° and we will find z

By using angle sum formula

x° + y° + z° = 180°

60° + 30° + z = 180°

z + 90° = 180°

z = 180° – 90°

z = 90°

Question 4: In a right-angled triangle one angle is 34°, find the other two angles?

Solution:

We know that in a right-angled triangle one angle is 90° and given angle is 34° and take an unknown angle as x°

Using angle sum formula 

90° + 34° + x° = 180°

124° + x° = 180°

x° = 180°-124°

x° = 56°

So other two angles are 90° and 56°

Question 5: Two angles of a triangle are 60° and 40° find the third angle.

Solution:

As we already know that sum of all the angles of a triangle is 180°

so consider x = 60° and y = 40° and we will find z

By using angle sum formula

x° + y° + z° = 180°

60° + 40° + z = 180°

z + 100° = 180°

z = 180° – 100°

z = 80°


Last Updated : 14 Feb, 2022
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