Why Is the Sum of Angles in a Triangle Always 180 Degrees?

A triangle has three sides and three angles, and the sum of three angles is 180 degrees. This is taught to everyone in school as a standard rule, but did it ever cross your thoughts why is the sum of a triangle 180°?  

Learn why with 98thPercentile and figure out how the standard rule is decoded. With 98thPercentile you can now understand the answer to all the why and how questions. Let's crack the shell open for why the sum of all three angles is 180°. 

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What is an Angle? 

An angle is formed when two straight lines meet at a common point. The common point is called the vertex of the angle, and it is measured in degrees. 

Understanding the sum of angles in a triangle is 180° with Pizza

To understand this explanation, you will require a pen, paper, scale, and protractor. But if you can imagine as you read this then it works too. Now picture this:  

• Draw a full pizza on paper. 

• Cut the pizza horizontally into half and use a protractor and measure the angle. (angle will be 180°) 

• Now cut the pizza vertically passing along the center which means 180° cut into half is 90° 

• You have 4 slices with 4 right angles. 

(Proven fact: One exterior angle = sum of two interior angles) 

•  Here you must particularly observe only one slice and consider it your slice 

• One side of the angle in the slice is already 90°. Let's say A1=90° 

• The angle of another pizza slice which is exterior to your pizza slice is also 90° 

• Now you are left with two interior angles of your pizza slice 

• According to the proof, the sum of the two interior angles is 90°. Let's say A2+A3= 90° 

Hence, A1+(A2+A3) = 90°+ 90° 

                               = 180° 

Therefore, the sum of angles in a triangle is 180° 

Theorem proof: 

• Draw a triangle XYZ with angles x,y, and z. 

• Draw a line AB passing through the vertex X, which is parallel to the side YZ.  

• Two angles will be formed, mark them as d and e.  

• Since AB is a transversal for the parallel lines AB and YZ, we have e = z (alternate interior angles)  

• Similarly, d = y.  

• Now x, y, and z must sum to 180° (angles on a straight line). Thus, p + a + q = 180°  

• Since e = z and d = y. Thus, x + y + z = 180°  

Therefore, the sum of the three angles x, y, and z is 180°. Hence proved. 

Are you curious about more interesting theorems now? Well then you are at the right place, 98thPercentile is an award-winning online Math program for students in grades 1-12 that can take your child a grade ahead, guaranteed. Join and discover the limitless possibilities of this fascinating subject. 

FAQ (Frequently Asked Questions) 

Q.1. What is the sum of all interior angles in a triangle?

Ans: The sum of angles in a triangle is 180°.

Q.2. Why is the sum of interior angles 180°? 

Ans: According to principles of Euclidean geometry the sum of interior angles is 180°.

Q.3. Is this rule the same for all triangles?

Ans: Yes, all the triangles have their sum of angle as 180°.

Q.4. How can verify the sum of angles in a triangle is 180°? 

Ans: Using a protractor to measure these angles in the triangle will give a practical verification.

Q.5. Does a right-angle triangle also follow the exterior angle theorem?

Ans: Yes, the right angle triangle also follows this theorem.

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