Triangle Angle Sum Theorem

Last updated on 29 November 2025

Table of contents

What is the Triangle Angle Sum Theorem?
Proof of the Triangle Angle Sum Theorem
Solved Examples on Triangle Angle Sum Theorem
FAQs on Triangle Angle Sum Theorem

Geometry, the age-old branch of mathematics, is a mesmerizing world of shapes and figures that has intrigued the human mind for centuries. Within this realm, the Triangle Angle Sum Theorem stands as a foundational principle that underpins the very essence of triangles. In this article, let's explore this fundamental theorem, uncovering its significance, implications, and applications in the realms of mathematics, science, and beyond.

What is the Triangle Angle Sum Theorem?

The Triangle Angle Sum Theorem, also known as the Triangle Sum Theorem, is a fundamental principle in geometry that relates to the interior angles of a triangle. It states that the sum of the all three interior angles of any triangle is always equal to 180°. This theorem applies to all types of triangles.

Proof of the Triangle Angle Sum Theorem

Given: In △ABC in the figure above,

$∠ BAC$ , $∠ ABC$ and $∠ ACB$ are interior angles.
$∠ BAC = a$ , $∠ ABC = b$ and $∠ ACB = c$

Construction:

Drawn line segment PQ parallel to side BC passing through the vertex A.
Angles $∠ PAB$ and $∠ QAC$ are formed by the line segment PQ.
Consider $∠ PAB = x$ and $∠ QAC = y$

To Prove: We have to prove that the sum of the angles $∠ ABC$ , $∠ BAC$ and $∠ ACB$ is 180°, i.e. $∠ ABC + ∠ BAC + ∠ ACB = 180 °$

Proof: Sides AB and AC are transversals for the parallel lines PQ and BC.
Hence, according to the alternate interior angles theorem, $\begin{aligned} (1) & x & = & b \\ (2) & y & = & c \end{aligned}$ The sum of the angles that are formed on a straight line at the same point is always 180°.
Therefore the sum of the angles $x$ , $a$ and $y$ is 180°, since these angles lie on the straight line PQ. $\begin{aligned} (3) & ∴ x + a + y = 180 ° \end{aligned}$ Put values from $(1)$ and $(2)$ in above equation $(3)$ . $\begin{aligned} ∴ b + a + c = 180 ° \\ ∴ ∠ ABC + ∠ BAC + ∠ ACB = 180 ° \end{aligned}$ Henced proved the triangle angle sum theorem.

Solved Examples on Triangle Angle Sum Theorem

Example 1: Find the value of

x

in the triangle shown below.

Given: In △PQR in the figure above, $∠ Q = 70 °$ $∠ P = ∠ R = x °$

Solution: By the Triangle Angle Sum Theorem, the sum of the interior angles of a triangle is 180°. $\begin{array}{lrcl} ∴ & ∠ P + ∠ Q + ∠ R & = & 180 ° \\ ∴ & x + 70 ° + x & = & 180 ° \\ ∴ & 2 x + 70 ° & = & 180 ° \\ ∴ & 2 x & = & 180 ° - 70 ° \\ ∴ & 2 x & = & 110 ° \\ ∴ & x & = & \frac{110 °}{2} \\ ∴ & x & = & 55 ° \end{array}$ Therefore the value of $x$ is $55 °$ .

FAQs on Triangle Angle Sum Theorem

What is the Triangle Angle Sum Theorem?
The Triangle Angle Sum Theorem states that the sum of the all three interior angles of a triangle is always 180°.
How do you use the Triangle Angle Sum Theorem?
To find an unknown angle in a triangle, you can subtract the sum of the known angles from 180°. For example, if two angles are known (let's say A and B), you can find the third angle C using the formula:
$C = 180 ° - (A + B)$
Does the Triangle Angle Sum Theorem apply to all types of triangles?
Yes, the theorem applies to all types of triangles, including equilateral, isosceles, scalene, acute-angled, obtuse-angled and right-angled triangles.
What happens if the angles do not sum to 180°?
If the angles of a shape do not sum to 180°, then the shape is not a triangle or the measurements are incorrect.
Can the Triangle Angle Sum Theorem be used in non-Euclidean geometry?
In non-Euclidean geometry, such as spherical or hyperbolic geometry, the sum of the angles in a triangle may not equal 180°. This theorem specifically applies to Euclidean geometry.
How can the Triangle Angle Sum Theorem help in solving geometric problems?
It provides a fundamental relationship between the angles in a triangle, allowing for the calculation of missing angles and understanding the properties of triangles in geometry.
Is there a formula for finding the angles of a triangle given the lengths of its sides?
Yes, the Law of Cosines can be used to find the angles of a triangle when the lengths of all three sides are known. For example, if the sides are $a$ , $b$ and $c$ , the angle C opposite side $c$ can be found using:
$C = c o s^{- 1} (\frac{a^{2} + b^{2} - c^{2}}{2 a b})$
Why is it important to know the Triangle Angle Sum Theorem?
Understanding this theorem is crucial in various fields such as mathematics, engineering, architecture, and physics, as it aids in solving problems related to triangles and shapes.
Can you give an example using the Triangle Angle Sum Theorem?
Certainly! If a triangle has two angles measuring 50° and 70°, you can find the third angle C:
$C = 180 ° - (50 ° + 70 °) = 180 ° - 120 ° = 60 °$
Are there any exceptions to the Triangle Angle Sum Theorem?
No, there are no exceptions in traditional Euclidean geometry; the theorem holds true for all triangles within that framework.

Triangle Angle Sum Theorem

What is the Triangle Angle Sum Theorem?

Proof of the Triangle Angle Sum Theorem

Solved Examples on Triangle Angle Sum Theorem

FAQs on Triangle Angle Sum Theorem

What is the Triangle Angle Sum Theorem?

How do you use the Triangle Angle Sum Theorem?

Does the Triangle Angle Sum Theorem apply to all types of triangles?

What happens if the angles do not sum to 180°?

Can the Triangle Angle Sum Theorem be used in non-Euclidean geometry?

How can the Triangle Angle Sum Theorem help in solving geometric problems?

Is there a formula for finding the angles of a triangle given the lengths of its sides?

Why is it important to know the Triangle Angle Sum Theorem?

Can you give an example using the Triangle Angle Sum Theorem?

Are there any exceptions to the Triangle Angle Sum Theorem?

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