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 is the 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 the angle sum rule? ›

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.

What is the sum property formula? ›

Formula for angle sum property:-

S = (n-2)*180° is the angle sum property formula for any polygon, where 'n' represents the number of sides in the polygon. The sum of the internal angles in a polygon may be determined using the number of triangles that can be constructed inside it, according to this polygon feature.

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 is the formula for angle sum? ›

Therefore, to find the sum of the interior angles of a polygon, we use the formula: Sum of interior angles = (n − 2) × 180° where 'n' = the number of sides of a polygon.

What is angle sum theorem math? ›

The triangle sum theorem (also known as the triangle angle sum theorem or angle sum theorem) states that the sum of the three interior angles of any triangle is always 180 degrees. An interior angle is an angle that is on the inside of a triangle.

What is the law of the sum of angles? ›

Theorem 1: Angle sum property of triangle states that the sum of interior angles of a triangle is 180°. Proof: Consider a ∆ABC, as shown in the figure below. To prove the above property of triangles, draw a line PQ parallel to the side BC of the given triangle.

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 a Euclidean space, the sum of angles of a triangle equals a straight angle (180 degrees, π radians, two right angles, or a half-turn). A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides.

What are angle properties? ›

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.

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