Angle Sum Property of a Triangle: Theorem - Definition With Proof

In this module, we will learn about an important property of a triangle which is the �Angle Sum Property of a Triangle�.

Activities To Show Angle Sum Property of a Triangle

1. Let us do an activity.

Draw a triangle ABC.

Cut on the three angles A, B and C.

Rearrange them as shown in the figure.

The three angles now constitute one angle.

This angle is a straight angle and so has measured 180�. Thus, the sum of the measures of the three angles of a triangle is 180�.

2. Let us do one more activity.

Draw any three triangles, say triangle ABC, triangle PQR and triangle XYZ.

Use your protractor and measure each of the angles of these triangles.

Tabulate your results as:

Name of ?	Measures of Angles	Sum of the Measures of the three Angles
?ABC	m ?A = 70�, m ?B = 80�, m ?C = 30�	m ?A + m ?B + m ?C = 70� + 80� + 30� = 180�
?PQR	m ?P = 50�, m ?Q = 40�, m ?R = 90�	m ?P + m ?Q + m ?R = 50� + 40� + 90� = 180�
?XYZ	m ?X = 100�, m ?Y = 40�, m ?Z = 40�	m ?X + m ?Y + m ?Z = 100� + 40� + 40� = 180�

Thus from both the activities, we can say that the sum of the measures of the three angles of a triangle is 180�.

Angle Sum Property of a Triangle Proof

We are now ready to give a formal justification of our assertion through logical argument.

Statement: The total measure of the three angles of a triangle is 180�.

To justify this let us use the exterior angle property of a triangle.

Given: ?1, ?2, ?3 are angles of ?ABC.

?4 is the exterior angle when BC is extended to D.

Let us justify our result.

?1 + ?2 = ?4 (By exterior angle property)

Adding ?3 to both the sides,

?1 + ?2 + ?3 = ?4 + ?3

But ?4 and ?3 form a linear pair so it is 180�.

Therefore,?1 + ?2 + ?3 = 180�