Angle Sum Property of a Triangle: Definition, Theorem, Formula (2024)

Home » Math Vocabulary » Angle Sum Property of a Triangle: Definition, Theorem, Examples

  • What is the Angle Sum Property of a Triangle?
  • Triangle Sum Theorem Proof
  • Exterior Angle Theorem
  • Solved Examples on Angle Sum Property of a Triangle
  • Practice Problems on Angle Sum Property of a Triangle
  • Frequently Asked Questions on Angle Sum Property of a Triangle

What is the Angle Sum Property of a Triangle?

The “angle sum property of a triangle theorem” (also known as the “triangle sum theorem” or “angle sum theorem”) states that the sum of the three interior angles of any triangle is always $180^{\circ}$.

Angle Sum Property of a Triangle: Definition, Theorem, Formula (1)

What is the angle sum theorem in geometry? In Euclidean geometry, any triangle whether it is a right triangle, an obtuse triangle, or an acute triangle all have interior angles that add up to 180 degrees.

Angle Sum Property of a Triangle: Definition, Theorem, Formula (2)Begin here

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Angle Sum Property of a Triangle Theorem

The angle sum property of a triangle theorem states that the sum of all three internal angles of a triangle is $180^{\circ}$. It is also known as the angle sum theorem or triangle sum theorem.

Angle Sum Property of a Triangle: Definition, Theorem, Formula (13)

According to the angle sum theorem, in the above ABC,

$m\angle A + m\angle B + m\angle C = 180^{\circ}$

Example: In PQR, $\angle P = 60^{\circ}, \angle Q = 70^{\circ}$

Angle Sum Property of a Triangle: Definition, Theorem, Formula (14)

According to the angle sum theorem, in the above triangle PQR,

$m\angle P + m\angle Q + m\angle R = 180^{\circ}$

$60^{\circ} + 70^{\circ} + m\angle R = 180^{\circ}$

$130^{\circ} + m\angle R = 180^{\circ}$

$m\angle R = 50^{\circ}$

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Triangle Sum Theorem Proof

In the figure given below, AB, BC, and CA represent three sides of triangle ABC. A, B, and C are the three vertices. $\angle A,\; \angle B,$ and $\angle C$ are the three interior angles of $\Delta ABC$.

Angle Sum Property of a Triangle: Definition, Theorem, Formula (25)

In ∆ABC, we have to prove that the sum of the angles $\angle A,\; \angle B,$ and $\angle C$ is $180^{\circ}$.

To prove: $m\angle A + m \angle B + m\angle C = 180^{\circ}$

Construction: Draw a line DE passing through the vertex B, which is parallel to the side AC.

At point B, two angles are formed, $\angle 1$ and $\angle 2$.

Since AB is a transversal for the parallel lines DE and AC, we have

$m\angle 1 = m\angle A$ (since the pair of alternate interior angles are equal)

Similarly, $m\angle 2 = m\angle C$.

Now, $m\angle 1,\; m\angle B,$ and $m\angle 2$ must add up to $180^{\circ}$ (angles on a straight line).

Thus, $\angle 1 + \angle B + \angle 2 = 180^{\circ}$… (I)

Since $\angle 1 = \angle A$ and $\angle 2 = \angle C$. … (II)

Substituting equation (II) in equation (I), we get

$m\angle A + m\angle B + m\angle C = 180^{\circ}$

Therefore, the sum of the three angles $\angle A,\; \angle B,$ and $\angle C$ is $180^{\circ}$.

Hence, the triangle sum theorem was proved.

Exterior Angle Theorem

The exterior angle theorem states that “an exterior angle of a triangle is equal to the sum of its two opposite interior angles.”

Angle Sum Property of a Triangle: Definition, Theorem, Formula (26)

In the above triangle, $\angle A,\; \angle B,$ and $\angle C$ are the interior angles of the triangle ABC, and $\angle 1,\; \angle 2,$ and $\angle 3$ are the exterior angle.

$m\angle A + m\angle B + m\angle C = 180^{\circ}$ (angle sum property) … (I)

Also, $m\angle A + m\angle 1 = 180^{\circ}$ (linear pair angle) … (II)

From (I) and (II), we get

$m\angle A + m\angle 1 = m\angle A + m\angle B + m\angle C$

$m\angle 1 = m\angle B + m\angle C$

Similarly, we can derive for other two exterior angles,∠2 and ∠3 which is given by:

$m\angle 2 = m\angle A + m\angle C$

$m\angle 3 = m\angle A + m\angle B$

In summary:

Angle Sum Property of a Triangle: Definition, Theorem, Formula (27)

Facts about Angle Sum Property of a Triangle

  • The theorem of angle sum property of triangles holds true for all types of triangles.
  • The sum of all exterior angles of a triangle is equal to $360^{\circ}$.
  • The sum of the lengths of any two sides of a triangle is always greater than the third side.
  • A rectangle can be divided into two right triangles by drawing a line from one corner to the opposite corner.
  • The study of the relationship between the sides and angles of triangles is known as trigonometry.
  • Due to their high strength, triangle shapes are frequently utilized in construction.
  • In North Carolina, there are three cities: Raleigh, Durham, and Chapel Hill, that are often referred to as the triangle.
  • The sum of all exterior angles of a triangle is equal to $360^{\circ}$.

Conclusion

In this article, we have learned all about the angle sum property of a triangle, exterior angle property of a triangle, proof of triangle sum theorem, and some important facts about triangles.

Let’s solve a few triangle angle sum theorem examples and practice problems.

Solved Examples on Angle Sum Property of a Triangle

  1. In a triangle, ABC, if $m\angle A = 60^{\circ},\; m\angle B = 40^{\circ}$, then find the measure of angle $\angle C$.

Solution:

In $\Delta ABC,\; \angle A = 60^{\circ}$ and $\angle B = 40^{\circ}$

We know that the sum of angles in a triangle is $180^{\circ}$.

$\Rightarrow m \angle A + \angle B + \angle C = 180^{\circ}$

$\Rightarrow 60^{\circ} + 40^{\circ} + \angle C = 180^{\circ}$

$\Rightarrow \angle C = 180^{\circ} \;−\; ( 60^{\circ} + 40^{\circ})$

$\Rightarrow \angle C = 180^{\circ} \;−\; 100^{\circ}$

$\therefore \angle C = 80^{\circ}$

  1. One of the acute angles in a right-angled triangle is $40^{\circ}$. Using the angle sum theorem, determine the other angle.

Solution:

Let $\Delta ABC$ be given a right-angled triangle which is right-angled at B.

$\therefore \angle B = 90^{\circ}$

$m\angle A = 40^{\circ}$ and we have to find out $m\angle C$.

We know that the sum of angles in a triangle is $180^{\circ}$.

$\Rightarrow m \angle A + m \angle B + m \angle C = 180^{\circ}$

$\Rightarrow 40^{\circ} + 90^{\circ} + m\angle C = 180^{\circ}$

$\Rightarrow m \angle C = 180^{\circ} \;−\; (40^{\circ} + 90^{\circ})$

$\Rightarrow m \angle C = 180^{\circ} \;−\; 130^{\circ}$

$\therefore m \angle C = 50^{\circ}$

$\therefore m \angle A = m \angle C = 50^{\circ}$

  1. The measures of interior angles of a triangle are $(2x\;−\;20)^{\circ}, (3x\;−\;10)^{\circ}$, and $2x^{\circ}$, find the values of all three angles of the triangle.

Solution:

We know that the sum of angles in a triangle is $180^{\circ}$

$\Rightarrow (2x\;−\;20)^{\circ} + (3x\;−\;10)^{\circ} + 2x^{\circ} = 180^{\circ}$

$\Rightarrow (2x\;−\;20 + 3x\;−\;10 + 2x)^{\circ} = 180^{\circ}$

$\Rightarrow 7x\;−\;30 = 180$

$\Rightarrow 7x = 180 + 30$

$\Rightarrow 7x = 210$

$\Rightarrow x = 2107= 30$

$\Rightarrow$ Angles are $40^{\circ},\; 80^{\circ}$ and $60^{\circ}$.

  1. Is it possible to construct a triangle with internal angles $43^{\circ},\; 49^{\circ},$ and $91^{\circ}$?

Solution:

Given, measurements of angles $43^{\circ},\; 49^{\circ},$ and $91^{\circ}$.

Here, $43^{\circ} + 49^{\circ} + 91^{\circ} = 183^{\circ} 180^{\circ}$

We know, that, the sum of angles in a triangle is $180^{\circ}$

Hence, it is not possible to construct a triangle with measurements of angles $43^{\circ},\; 49^{\circ},$ and $91^{\circ}$.

  1. In the figure given below, determine the value of “x.”
Angle Sum Property of a Triangle: Definition, Theorem, Formula (28)

Solution:

In the above figure, $\angle x$ is an exterior angle and $\angle A = 55^{\circ}$ and $\angle B = 47^{\circ}$ are given as interior angles.

According to the exterior angle property,

an exterior angle of a triangle is equal to the sum of its two interior opposite angles.

So, $\angle x = \angle A + \angle B = 55^{\circ} + 47^{\circ} = 102^{\circ}$

Hence, $x = 102^{\circ}$

Practice Problems on Angle Sum Property of a Triangle

1

If all angles of a triangle are congruent, then each angle measures ___.

$80^{\circ}$

$40^{\circ}$

$60^{\circ}$

$50^{\circ}$

CorrectIncorrect

Correct answer is: $60^{\circ}$
Sum of all angles of a triangle is $180^{\circ}$. If all angles of a triangle are congruent, then we have
$3x = 180^{\circ}$. Thus, $x = 60^{\circ}$.

2

What is the sum of all interior angles of a triangle?

$60^{\circ}$

$90^{\circ}$

$120^{\circ}$

$180^{\circ}$

CorrectIncorrect

Correct answer is: $180^{\circ}$
According to the angle sum theorem the sum of all three internal angles of a triangle is $180^{\circ}$.

3

Two angles of a triangle measure $30^{\circ}$ and $60^{\circ}$. The measure of the third angle is _____.

$30^{\circ}$

$90^{\circ}$

$130^{\circ}$

$180^{\circ}$

CorrectIncorrect

Correct answer is: $90^{\circ}$
We know that the sum of angles in a triangle is $180^{\circ}$.
Thus, the measure of the third angle $= 180^{\circ} \;−\; (30^{\circ} + 60^{\circ}) = 90^{\circ}$

4

Find x.

Angle Sum Property of a Triangle: Definition, Theorem, Formula (29)

$100^{\circ}$

$109^{\circ}$

$105^{\circ}$

$104^{\circ}$

CorrectIncorrect

Correct answer is: $109^{\circ}$
Exterior angle property of a triangle states that “an exterior angle of a triangle is equal to the sum of its two interior opposite angles.”Thus, $x = 109^{\circ}$

5

What is the sum of all exterior angles of a triangle?

$90^{\circ}$

$180^{\circ}$

$270^{\circ}$

$360^{\circ}$

CorrectIncorrect

Correct answer is: $360^{\circ}$
Each exterior angle is the sum of its two opposite interior angles. Thus, the sum of all the exterior angles is two times the sum of all interior angles. The sum of all exterior angles of a triangle is equal to $360^{\circ}$.

Frequently Asked Questions on Angle Sum Property of a Triangle

The angle sum theorem states that the sum of all three internal angles of a triangle is 180°. Whereas the exterior angle theorem states that an exterior angle of a triangle is equal to the sum of its two interior opposite angles.

Yes, we can define the angle sum property for a quadrilateral. According to the angle sum property of a quadrilateral, the sum of all four interior angles is 360 degrees.

The sum of all exterior angles of any polygon is equal to $360^{\circ}$. Hence the sum of all exterior angles of a quadrilateral is also equal to $360^{\circ}$

We know that the sum of angles in a triangle is $180^{\circ}$. For $\Delta ABC$, the formula for the angle sum property of a triangle is $\angle A + \angle B + \angle C = 180^{\circ}$.

According to the angle sum theorem for any polygon, the sum of all interior angles is equal to$(n − 2) \times 180^{\circ}$, where n is the total number of sides of the polygon.

Angle Sum Property of a Triangle: Definition, Theorem, Formula (2024)

FAQs

Angle Sum Property of a Triangle: Definition, Theorem, Formula? ›

The angle sum property of a triangle says that the sum of its interior angles is equal to 180°. Whether a triangle is an acute, obtuse, or a right triangle, the sum of the angles will always be 180°. This can be represented as follows: In a triangle ABC, ∠A + ∠B + ∠C = 180°.

What is the formula for the angle sum property of a triangle? ›

This property states that the sum of all the interior angles of a triangle is 180°. If the triangle is ∆ABC, the angle sum property formula is ∠A+∠B+∠C = 180°.

What is the answer of triangle angle sum theorem? ›

Answer: The sum of the three angles of a triangle is always 180 degrees. To find the measure of the third angle, find the sum of the other two angles and subtract that sum from 180.

What is the theorem for the angles of a triangle? ›

Theorem 1: The sum of all the three interior angles of a triangle is 180 degrees. Theorem 2: The base angles of an isosceles triangle are congruent. The angles opposite to equal sides of an isosceles triangle are also equal in measure.

What is the property theorem of a triangle? ›

The properties of the triangle are: The sum of all the angles of a triangle (of all types) is equal to 180°. The sum of the length of the two sides of a triangle is greater than the length of the third side. In the same way, the difference between the two sides of a triangle is less than the length of the third side.

What is the formula for angle addition property? ›

The Angle Addition Postulate states that the sum of two adjacent angle measures will equal the angle measure of the larger angle that they form together. The formula for the postulate is that if D is in the interior of ∠ ABC then ∠ ABD + ∠ DBC = ∠ ABC. Adjacent angles are two angles that share a common ray.

What are the angle sum formulas? ›

The angle sum identities take two different formulas: sin(A+B) = sinAcosB + cosAsinB. cos(A+B) = cosAcosB − sinAsinB.

What is the angle formula? ›

Angles Formulas at the center of a circle can be expressed as, Central angle, θ = (Arc length × 360º)/(2πr) degrees or Central angle, θ = Arc length/r radians, where r is the radius of the circle.

What is the equation for the triangle theorem? ›

The Pythagorean Theorem describes the relationship among the three sides of a right triangle. In any right triangle, the sum of the areas of the squares formed on the legs of the triangle equals the area of the square formed on the hypotenuse: a2 + b2 = c2.

What is the triangle angle sum theorem review? ›

All three angles in any triangle always add up to 180 degrees. So if you only have two of the angles with you, just add them together, and then subtract the sum from 180. EX: A Triangle has three angles A, B, and C.

What makes a unique triangle? ›

A "unique triangle" is a triangle for which there is no other triangle that has the same dimensions and shape. Duplicates of such triangles will all be congruent to one another. The word "unique" means "being the only one of its kind".

What is the formula for the triangle sum theorem? ›

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°.

What is the definition and theorem of triangle? ›

Theorem 1: The total of the three interior angles in any triangle is 180 degrees. Theorem 2: When a triangle side is constructed, the exterior angle formed is equal to the sum of the interior opposite angles. Theorem 3: The base angles of an isosceles triangle are equivalent.

What is the angle theorem rule? ›

A polygon is equiangular if the interior angles are congruent to each other and equilateral if the edges are congruent. An equiangular and equilateral polygon is called a regular polygon. The interior angle theorem states that the sum of the interior angles of a polygon with n vertices is S n = 180 ( n − 2 ) .

What is the SSS property of triangles? ›

SSS (Side-Side-Side)

If all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent by SSS rule. In the above-given figure, AB= PQ, BC = QR and AC=PR, hence Δ ABC ≅ Δ PQR.

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