Angle Sum Theorem
The angle 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 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 angle 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 Angle Sum Theorem?|
|2.||Exterior Angle Sum Theorem|
|3.||Polygon Angle Sum Theorem|
|4.||Angle Sum Theorem Examples|
|5.||FAQs on Angle Sum Theorem|
What Is the Angle Sum Theorem?
A triangle is a two-dimensional closed figure formed by three line segments and consists of the interior as well as exterior angles. As per the angle 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 above-given 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 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°.
Angle Sum Theorem Proof
Consider a triangle ABC. We have to show that the sum of the angles a, b, and c is 180°.
- 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°.
Exterior Angle Sum Theorem
A very important consequence of the angle sum property of triangles 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, c + α = 180° (angle sum property) _______ (2)
- From (1) and (2): a + b = 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 above-given polygon, we can observe that in this 5-sided 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(n-2)/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.
Related Articles on Angle Sum Theorem
Check out the following pages related to the angle sum theorem
Important Notes on Angle 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°.
- Angle 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°.
Angle Sum Theorem Examples
Example 1: One of the acute angles of a right-angled triangle is 45°. Find the other angle using the angle sum theorem. Identify the type of triangle thus formed.
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°)
Since two angles measure the same, it is an isosceles triangle.
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)°.
Given: angles of a triangle y°, (y + 20)° and (2y + 40)°.
As per the angle 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°.
Therefore, the three angles of the given triangle are 30°, 50°, and 100°.
FAQs on Angle Sum Theorem
What Is the Angle Sum Theorem in Geometry?
As per the angle 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 angle 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 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 two-dimensional 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°.