Angle Sum Property | Theorem | Proof | Examples- Cuemath (2024)

The angle sum property of a triangle states that the sum of the angles of a triangle is equal to 180º. A triangle has three sides and three angles, one at each vertex. Whether a triangle is an acute, obtuse, or a right triangle, the sum of its interior angles is always 180º.

The angle sum property of a triangle is one of the most frequently used properties in geometry. This property is mostly used to calculate the unknown angles.

1.What is the Angle Sum Property?
2.Angle Sum Property Formula
3.Proof of the Angle Sum Property
4.FAQs on Angle Sum Property

What is the Angle Sum Property?

According to the angle sum property of a triangle, the sum of all three interior angles of a triangle is 180 degrees. A triangle is a closed figure formed by three line segments, consisting of interior as well as exterior angles. The angle sum property is used to find the measure of an unknown interior angle when the values of the other two angles are known. Observe the following figure to understand the property.

Angle Sum Property | Theorem | Proof | Examples- Cuemath (1)

Angle Sum Property Formula

The angle sum property formula for any polygon is expressed as, S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. This property of a polygon states that the sum of the interior angles in a polygon can be found with the help of the number of triangles that can be formed inside it. These triangles are formed by drawing diagonals from a single vertex. However, to make things easier, this can be calculated by a simple formula, which says that if a polygon has 'n' sides, there will be (n - 2) triangles inside it. For example, let us take a decagon that has 10 sides and apply the formula. We get, S = (n − 2) × 180°, S = (10 − 2) × 180° = 10 × 180° = 1800°. Therefore, according to the angle sum property of a decagon, the sum of its interior angles is always 1800°. Similarly, the same formula can be applied to other polygons. The angle sum property is mostly used to find the unknown angles of a polygon.

Proof of the Angle Sum Property

Let's have a look at the proof of the angle sum property of the triangle. The steps for proving the angle sum property of a triangle are listed below:

  • Step 1: Draw a line PQ that passes through the vertex A and is parallel to side BC of the triangle ABC.
  • Step 2: We know that the sum of the angles on a straight line is equal to 180°. In other words, ∠PAB + ∠BAC + ∠QAC = 180°, which gives, Equation 1: ∠PAB + ∠BAC + ∠QAC = 180°
  • Step 3: Now, since line PQ is parallel to BC. ∠PAB = ∠ABC and ∠QAC = ∠ACB. (Interior alternate angles), which gives, Equation 2: ∠PAB = ∠ABC, and Equation 3: ∠QAC = ∠ACB
  • Step 4: Substitute ∠PAB and ∠QAC with ∠ABC and ∠ACB respectively, in Equation 1 as shown below.

Equation 1: ∠PAB + ∠BAC + ∠QAC = 180°. Thus we get, ∠ABC + ∠BAC + ∠ACB = 180°

Angle Sum Property | Theorem | Proof | Examples- Cuemath (2)

Hence proved, in triangle ABC, ∠ABC + ∠BAC + ∠ACB = 180°. Thus, the sum of the interior angles of a triangle is equal to 180°.

Important Points

The following points should be remembered while solving questions related to the angle sum property.

  • The angle sum property formula for any polygon is expressed as, S = ( n − 2) × 180°, where 'n' represents the number of sides in the polygon.
  • The angle sum property of a polygon states that the sum of the interior angles in a polygon can be found with the help of the number of triangles that can be formed inside it.
  • The sum of the interior angles of a triangle is always 180°.

Impoprtant Topics

  • Exterior Angle Theorem
  • Angles
  • Triangles

FAQs on Angle Sum Property

What is the Angle Sum Property of a Polygon?

The angle sum property of a polygon states that the sum of all the angles in a polygon can be found with the help of the number of triangles that can be formed in it. These triangles are formed by drawing diagonals from a single vertex. However, this can be calculated by a simple formula, which says that if a polygon has 'n' sides, there will be (n - 2) triangles inside it. The sum of the interior angles of a polygon can be calculated with the formula: S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. For example, if we take a quadrilateral and apply the formula using n = 4, we get: S = (n − 2) × 180°, S = (4 − 2) × 180° = 2 × 180° = 360°. Therefore, according to the angle sum property of a quadrilateral, the sum of its interior angles is always 360°. Similarly, the same formula can be applied to other polygons. The angle sum property is mostly used to find the unknown angles of a polygon.

What is the Angle Sum Property of a Triangle?

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 Angle Sum Property of a Hexagon?

According to the angle sum property of a hexagon, the sum of all the interior angles of a hexagon is 720°. In order to find the sum of the interior angles of a hexagon, we multiply the number of triangles in it by 180°. This is expressed by the formula: S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. In this case, 'n' = 6. Therefore, the sum of the interior angles of a hexagon = S = (n − 2) × 180° = (6 − 2) × 180° = 4 × 180° = 720°.

What is the Angle Sum Property of a Quadrilateral?

According to the angle sum property of a quadrilateral, the sum of all its four interior angles is 360°. This can be calculated by the formula, S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. In this case, 'n' = 4. Therefore, the sum of the interior angles of a quadrilateral = S = (4 − 2) × 180° = (4 − 2) × 180° = 2 × 180° = 360°.

What is the Exterior Angle Sum Property of a Triangle?

The exterior angle theorem says that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.

What is the Formula of Angle Sum Property?

The formula for the angle sum property is, S = ( n − 2) × 180°, where 'n' represents the number of sides in the polygon. For example, if we want to find the sum of the interior angles of an octagon, in this case, 'n' = 8. Therefore, we will substitute the value of 'n' in the formula, and the sum of the interior angles of an octagon = S = (n − 2) × 180° = (8 − 2) × 180° = 6 × 180° = 1080°.

What is the Angle Sum Property of a Pentagon?

As per the angle sum property of a pentagon, the sum of all the interior angles of a pentagon is 540°. In order to find the sum of the interior angles of a pentagon, we multiply the number of triangles in it by 180°. This is expressed by the formula: S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. In this case, 'n' = 5. Therefore, the sum of the interior angles of a pentagon = S = (n − 2) × 180° = (5 − 2) × 180° = 3 × 180° = 540°.

How to Find the Third Angle in a Triangle?

We know that the sum of the angles of a triangle is always 180°. Therefore, if we know the two angles of a triangle, and we need to find its third angle, we use the angle sum property. We add the two known angles and subtract their sum from 180° to get the measure of the third angle. For example, if two angles of a triangle are 70° and 60°, we will add these, 70 + 60 = 130°, and we will subtract it from 180°, which is the sum of the angles of a triangle. So, the third angle = 180° - 130° = 50°.

How to Find the Exterior Angle of a Polygon?

The exterior angle of a polygon is the angle formed between any side of a polygon and a line that is extended from the adjacent side. In order to find the measure of an exterior angle of a regular polygon, we divide 360 by the number of sides 'n' of the given polygon. For example, in a regular hexagon, where 'n' = 6, each exterior angle will be 60° because 360 ÷ n = 360 ÷ 6 = 60°. It should be noted that the corresponding interior and exterior angles are supplementary and the exterior angles of a regular polygon are equal in measure.

Angle Sum Property | Theorem | Proof | Examples- Cuemath (2024)

FAQs

Angle Sum Property | Theorem | Proof | Examples- Cuemath? ›

For example, if we take a quadrilateral and apply the formula using n = 4, we get: S = (n − 2) × 180°, S = (4 − 2) × 180° = 2 × 180° = 360°. Therefore, according to the angle sum property of a quadrilateral, the sum of its interior angles is always 360°. Similarly, the same formula can be applied to other polygons.

What is the angle sum property? ›

Angle Sum Property of a Triangle

It is also known as the interior angle 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 are the 7 properties of a triangle? ›

Properties
  • A triangle has three sides and three angles.
  • The sum of the angles of a triangle is always 180 degrees.
  • The exterior angles of a triangle always add up to 360 degrees.
  • The sum of consecutive interior and exterior angle is supplementary.

What is angle sum property class8? ›

According to the angle sum property of a Quadrilateral, the sum of all the four interior angles is 360 degrees. Proof: In the quadrilateral ABCD, ∠ABC, ∠BCD, ∠CDA, and ∠DAB are the internal angles.

What is the angle sum formula? ›

The sum of the interior angles of a given polygon = (n − 2) × 180°, where n = the number of sides of the polygon.

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 is an example of an angle property? ›

The angle properties of lines are: Vertically opposite angles are equal, for example a = d, b = c. Adjacent angles add to 180o, for example a + b = 180o, a + c = 180. Corresponding angles are equal, for example a = e, b = f, c = g, d= h.

What is the unique property 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 are Pythagoras properties of triangles? ›

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°.

What are the 5 triangle congruence properties? ›

The triangle congruence criteria are:
  • SSS (Side-Side-Side)
  • SAS (Side-Angle-Side)
  • ASA (Angle-Side-Angle)
  • AAS (Angle-Angle-Side)
  • HL (Hypotenuse-Leg, right triangle only)

What is the triangle sum rule? ›

The Triangle Sum Theorem states that if you add all three interior angles, those are the angles inside the triangle, they would add up to 180 degrees. It is easy to remember that we add the three angle measurements to get 180 degrees because of the word sum in the name of the theorem.

What is exterior angle sum property? ›

The properties of the exterior angle is given as follows: The exterior angle of a given triangle equals the sum of the opposite interior angles of that triangle. If an equivalent angle is taken at each vertex of the triangle, the exterior angles add to 360° in all the cases.

What is the special angle property? ›

Special angles are those found on the unit circle. Special angles are at 0 degrees, 30 degrees, 45 degrees, 60 degrees, and 90 degrees.

Is sin odd or even? ›

Sine is an odd function, and cosine is an even function. You may not have come across these adjectives “odd” and “even” when applied to functions, but it's important to know them. A function f is said to be an odd function if for any number x, f(–x) = –f(x).

What are the 12 trigonometric identities? ›

Sum and Difference of Angles Trigonometric Identities
  • sin(α+β)=sin(α).cos(β)+cos(α).sin(β)
  • sin(α–β)=sinα.cosβ–cosα.sinβ
  • cos(α+β)=cosα.cosβ–sinα.sinβ
  • cos(α–β)=cosα.cosβ+sinα.sinβ
  • tan. ⁡ ( α + β ) = tan ⁡ ⁡ β 1 – tan ⁡ α . tan. ⁡ β
  • tan. ⁡ ( α – β ) = tan ⁡ ⁡ β 1 + tan ⁡ α . tan. ⁡

Is tan sin cos or cos sin? ›

As we know, tan is the ratio of sin and cos, such as tan θ = sin θ/cos θ. Thus, we can get the values of tan ratio for the specific angles.

What is the property of sum? ›

The 4 main properties of addition are commutative, associative, distributive, and additive identity. Commutative refers that the result obtained from addition is still the same if the order changes. Associative property denotes that the pattern of summing up 3 numbers does not influence the result.

Is angle sum property of a triangle 180 degree? ›

In the given triangle, ∆ABC, AB, BC, and CA represent three sides. A, B and C are the three vertices and ∠ABC, ∠BCA and ∠CAB are three interior angles of ∆ABC. Theorem 1: Angle sum property of triangle states that the sum of interior angles of a triangle is 180°.

What is the property of an angle? ›

Properties of Angles

Important properties of the angle are: For one side of a straight line, the sum of all the angles always measures 180 degrees. The sum of all angles always measures 360 degrees around a point. An angle is a figure where, from a common position, two rays appear.

What is the angle angle side property? ›

The angle-angle-side theorem, or AAS, tells us that if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent.

References

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