Triangle Sum Theorem (Angle Sum Theorem)
The triangle sum theorem states that the sum of all the interior angles of a triangle is 180 degrees. In a Euclidean space, the sum of the measure of the interior angles of a triangle sum up to 180 degrees, be it an acute, obtuse, or a right triangle which is the direct result of the triangle sum theorem, also known as the angle sum theorem of the triangle. A triangle is the smallest polygon having three sides and three interior angles, one at each vertex, bounded by a pair of adjacent sides.
In geometry, the triangle sum theorem has varied applications as it gives important results while solving problems involving triangles and other polygons. In this article, we will discuss the angle sum theorem and the exterior angle theorem of a triangle with its statement, proof, and examples.
1.  What Is the Triangle Sum Theorem? 
2.  Triangle Sum Theorem Formula 
3.  Triangle Sum Theorem Proof 
4.  Exterior Angle Sum Theorem 
5.  Polygon Angle Sum Theorem 
6.  FAQs on Angle Sum Theorem 
What Is the Triangle Sum Theorem?
A triangle is a twodimensional closed figure formed by three line segments and consists of the interior as well as exterior angles. As per the triangle sum theorem, the sum of all the angles (interior) of a triangle is 180 degrees, and the measure of the exterior angle of a triangle equals the sum of its two opposite interior angles.
Consider a triangle ABC as shown below:
From the abovegiven figure, we can notice that all three angles of the triangle when rearranged, constitute one straight angle. So, ∠A + ∠B + ∠C = 180°. Thus, in the given triangle ABC, ∠A + ∠B + ∠C = 180°. Let's consider an example to understand this theorem. Consider a triangle PQR such that, ∠P = 38° and ∠Q = 134°. Calculate ∠R. As per the triangle angle sum theorem, ∠P + ∠Q + ∠R = 180°
⇒ 38° + 134° + ∠R = 180°
⇒ 172° + ∠R = 180°
⇒ ∠R = 180° – 172°
Therefore, ∠R = 8°
Angle Sum Theorem Statement
Statement: The angle sum theorem states that the sum of all the interior angles of a triangle is 180 degrees.
Triangle Sum Theorem Formula
The sum of the interior angles in a triangle is supplementary. In other words, the sum of the measure of the interior angles of a triangle equals 180°. So, the formula of the triangle sum theorem can be written as, for a triangle ABC, we have ∠A + ∠B + ∠C = 180°.
Triangle Sum Theorem Proof
Consider a triangle ABC. We have to show that the sum of the angles a, b, and c is 180°.
Proof:
 Draw a line DE passing through the vertex A, which is parallel to the side BC.
 Two angles will be formed, mark them as p and q.
 Since AB is a transversal for the parallel lines DE and BC, we have p = b (alternate interior angles)
 Similarly, q = c.
 Now, p, a, and q must sum to 180° (angles on a straight line). Thus, p + a + q = 180°
 Since p = b and q = c. Thus, a + b + c = 180°
Therefore, the sum of the three angles a, b, and c is 180°. Hence, we have proved the triangle sum theorem.
Exterior Angle Sum Theorem
A very important consequence of the triangle sum theorem is the exterior angle theorem which states that "an exterior angle of a triangle is equal to the sum of its two interior opposite angles."
 In the above triangle, a, b, and c are interior angles of the triangle ABC, and α is the exterior angle.
 a + b + c = 180° (angle sum property) _______ (1)
 Also, b + α = 180° (Linear Pair) _______ (2)
 From (1) and (2): a + c = α
Polygon Angle Sum Theorem
The polygon exterior angle sum theorem states that "the sum of all exterior angles of a convex polygon is equal to 360°'. Let's consider the polygon given below.
In the abovegiven polygon, we can observe that in this 5sided polygon, the sum of all exterior angles is 360° by polygon angle sum theorem. The number of interior angles is equal to the number of sides. The measure of an interior angle of a regular polygon can be calculated using the formula, Interior angle = 180º(n2)/n, where n is the number of sides. Each exterior angle of a regular polygon is equal and the sum of the exterior angles of a polygon is 360°. An exterior angle can be calculated using the formula, Exterior Angle = 360º/n, where n is the number of sides.
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Important Notes on Triangle Sum Theorem
Here is a list of a few important points on the angle sum theorem.
 The sum of all interior angles of a triangle is equal to 180°.
 Triangle sum theorem holds for all types of triangles.
 The sum of all exterior angles of a triangle is equal to 360°.
 The sum of all exterior angles of a convex polygon is equal to 360°.
Triangle Sum Theorem Examples

Example 1: One of the acute angles of a rightangled triangle is 45°. Find the other angle using the triangle sum theorem. Identify the type of triangle thus formed.
Solution:
Given, ∠1 = 90° (right triangle) and ∠2_{ }= 45°.
We know that the sum of the angles of a triangle adds up to 180°.
Therefore, ∠3 = 180°  (90° + 45°)
= 45°.
Since two angles measure the same, it is an isosceles triangle.
Answer: Therefore, the given triangle ABC is an isosceles triangle.

Example 2: Using the angle sum theorem, calculate the value of y for a triangle whose angles are y°, (y + 20)°, and (2y + 40)°.
Solution:
Given: angles of a triangle y°, (y + 20)° and (2y + 40)°.
As per the triangle sum theorem, the sum of interior angles = 180°
y° + (y + 20)° + (2y + 40)° = 180°
Now, let's simplify.
y + y + 2y + 20° + 40° = 180°
4y + 60° = 180°
4y = 180° – 60°4y = 120°
y = 120°/4
y = 30°
Thus, the angles of the given triangle are as follows:
y = 30°.
(y + 20)° = 30° + 20° = 50°.
(2y + 40)° = 2 × 30° + 40° = 60° + 40° = 100°.
Answer: Therefore, the three angles of the given triangle are 30°, 50°, and 100°.

Example 3: The three angles of a triangle are 35°, 67°, and 100°. Is the statement true? (Use Triangle Sum Theorem)
Solution: To identify if the statement is true, let us use the triangle sum theorem and add the angles.
35° + 67° + 100° = 202° ≠ 180°
According to the angle sum theorem, the sum of interior angles of a triangle is 180 degrees. So, the given statement is not true.
Answer: The statement is not true.
FAQs on Triangle Sum Theorem
What Is the Triangle Sum Theorem in Geometry?
As per the triangle sum theorem, in any triangle, the sum of the three angles is 180°. There are different types of triangles in mathematics as per their sides and angles. All of these triangles have three angles and they all follow the triangle sum theorem.
What Is the Formula for Triangle Sum Theorem?
Consider a triangle ABC. In this given triangle ABC, ∠a + ∠b + ∠c = 180°. This is the formula for the angle sum theorem. The sum of the interior angles in a triangle is supplementary.
What Is the Angle Sum Formula for Polygons?
We have the formula to find the sum of interior angles of a polygon. For this, we need to multiply the number of triangles in the polygon by the angle of 180°. The formula that is used for finding the sum of interior angles is (n − 2) × 180°, where n is the number of sides.
What Is the Exterior Angle Sum Theorem?
The polygon exterior angle sum theorem states that the sum of all exterior angles of a convex polygon is equal to 360°.
What Does the Triangle Sum Theorem State?
The angle sum theorem states that the sum of all the interior angles of a triangle is 180 degrees.
How to Prove the Triangle Sum Theorem?
We can prove the triangle sum theorem by making a line passing through one of the vertices of the triangle and parallel to the opposite side. Then, we can use the parallel lines and transversal results, and the sum of angles of on a straight line property to prove the triangle sum theorem.
What Is the Angle Sum Theorem for Quadrilaterals?
Each of the quadrilaterals has four sides, four vertices, four interior angles, and two diagonals. The angle sum theorem of quadrilateral states that the sum of all interior angles is 360°. As per the angle sum theorem for quadrilaterals, the sum of all measures of the interior angles of the quadrilateral is 360°.
What Is Polygon Angle Sum Theorem?
Polygons are twodimensional figures with more than 3 sides. As per the polygon angle sum theorem, the sum of the interior angle measures of a polygon depends on the number of sides it has. Also, by dividing a polygon with the number of sides it has, let it be n sides into (n – 2) triangles, it can be shown that the sum of the interior angle of any polygon is a multiple of 180°.
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