Quadrilateral Angles Sum Property - Theorem and Proof (2024)

Before talking about the quadrilateralsangle sum property, let us recall what angles and quadrilateral is. The angle is formed when two line segment joins at a single point. An angle is measured in degrees (°). Quadrilateral angles are the angles formed inside the shape of a quadrilateral. The quadrilateral is four-sided polygon which can have or not have equal sides. It is a closed figure in two-dimension and has non-curved sides. A quadrilateral is a polygon which has 4 vertices and 4 sides enclosing 4 angles andthe sum of all the angles is 360°. When we draw a draw the diagonals to the quadrilateral, it forms two triangles. Both these triangles have an angle sum of 180°. Therefore, the total angle sum of the quadrilateral is 360°. Angle sum is one of the properties of quadrilaterals. In this article, w will learn the rules of angle sum property.

Quadrilateral	Area Of Quadrilateral
Construction Of Quadrilaterals	Types Of Quadrilaterals

Angle Sum Property of a Quadrilateral

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.
AC is a diagonal
AC divides the quadrilateral into two triangles, ∆ABC and ∆ADC

We have learned that the sum of internal angles of a quadrilateral is 360°, that is, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.

let’s prove that the sum of all the four angles of a quadrilateral is 360 degrees.

We know that the sum of angles in a triangle is 180°.
Now consider triangle ADC,

∠D + ∠DAC + ∠DCA = 180° (Sum of angles in a triangle)

Now consider triangle ABC,

∠B + ∠BAC + ∠BCA = 180° (Sum of angles in a triangle)

On adding both the equations obtained above we have,

(∠D + ∠DAC + ∠DCA) + (∠B + ∠BAC + ∠BCA) = 180° + 180°

Quadrilateral Angles

A quadrilateral has 4 angles. The sum of its interior angles is 360 degrees. We can find the angles of a quadrilateral if we know 3 angles or 2 angles or 1 angle and 4 lengths of the quadrilateral. In the image given below, a Trapezoid (also a type of Quadrilateral) is shown.

The sum of all the angles∠A +∠B +∠C +∠D = 360°

In the case of square and rectangle, the value of all the angles is 90 degrees. Hence,

∠A = ∠B = ∠C = ∠D = 90°

A quadrilateral, in general, has sides of different lengths and angles of different measures. However, squares, rectangles, etc. are special types of quadrilaterals with some of their sides and angles being equal.

Do the Opposite side in a Quadrilateral equals 180 Degrees?

There is no relationship between the opposite side and the angle measures of a quadrilateral. To prove this, the scalene trapezium has the side length of different measure, which does not have opposite angles of 180 degrees. But in case of some cyclic quadrilateral, such as square, isosceles trapezium, rectangle, the opposite angles are supplementary angles. It means that the angles add up to 180 degrees. One pair of opposite quadrilateral angles are equal in the kite and two pair of the opposite angles are equal in the quadrilateral such as rhombus and parallelogram. It means that the sum of the quadrilateral angles is equal to 360 degrees, but it is not necessary that the opposite angles in the quadrilateral should be of 180 degrees.

Types of Quadrilaterals

There are basically five types of quadrilaterals. They are;

Parallelogram: Which has opposite sides as equal and parallel to each other.
Rectangle: Which has equal opposite sides but all the angles are at 90 degrees.
Square: Which all its four sides as equal and angles at 90 degrees.
Rhombus: Its a parallelogram with all its sides as equal and its diagonals bisects each other at 90 degrees.
Trapezium: Which has only one pair of sides as parallel and the sides may not be equal to each other.

Example

1. Find the fourth angle of a quadrilateral whose angles are 90°, 45° and 60°.

Solution: By the angle sum property we know;

Sum of all the interior angles of a quadrilateral = 360°

Let the unknown angle be x

So,

90° + 45° + 60° + x = 360°

195° + x = 360°

x = 360° – 195°

x = 165°

To learn more about quadrilaterals and their properties, download BYJU’S-The Learning App.

FAQs

Quadrilateral Angles Sum Property - Theorem and Proof? ›

A quadrilateral is a polygon which has 4 vertices and 4 sides enclosing 4 angles and the sum of all the angles is 360°. When we draw a draw the diagonals to the quadrilateral, it forms two triangles. Both these triangles have an angle sum of 180°. Therefore, the total angle sum of the quadrilateral is 360°.

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What is the proof of angle sum property of a quadrilateral? ›

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°.

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How do you prove the angle sum property? ›

To prove the above property of triangles, draw a line PQ parallel to the side BC of the given triangle. Thus, the sum of the interior angles of a triangle is 180°.

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How to prove that in a quadrilateral the sum of all the exterior angles is 360 degree? ›

In a concave quadrilateral, one of the angles is more than 180° and one of the diagonals is out of the region of triangle. Let sum of all exterior angles be 'E', and sum of all interior angles be 'I'. E = n × 180° - (n -2) × 180°. Hence, The sum of all the exterior angles of a polygon is 360° .

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What are the rules for the angles of a quadrilateral? ›

The angles that lie inside a quadrilateral are called its interior angles. The sum of the interior angles of a quadrilateral is 360°. This helps in calculating the unknown angles of a quadrilateral. In case if the quadrilateral is a square or a rectangle, then we know that all its interior angles are 90° each.

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How do you prove that ABCD is a quadrilateral? ›

We can say that a quadrilateral is a closed figure with four sides : e.g. ABCD is a quadrilateral which has four sides AB, BC, CD and DA, four angles ∠A,∠B,∠C and ∠D and four vertices A, B, C and D and also has two diagonals AC and BD. i.e. A quadrilateral has four sides, four angles, four vertices and two diagonals.

What is the quadrilateral congruence theorem? ›

If they have a side together with the adjacent angles respectively congruent, then the quadrilaterals are congruent. Proof. In order to show that Q = (A, B, C, D) and Q/ = (A/,B/,C/,D/) are congruent, we may suppose that they have AB = A/B/, ˆA = ˆA/ and ˆB = ˆB/.

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How many theorems are in quadrilateral? ›

1st Theorem- The diagonal divides the parallelogram into two congruent triangles. 2nd Theorem- The opposite side of a parallelogram are equal. 3rd Theorem- The quadrilaterals in which each pair of opposite sides are equal are called parallelogram. 4th Theorem- The opposite angles of a parallelogram are equal.

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What is angle sum property of all angles? ›

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.

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How do you prove the angle angle side theorem? ›

In order to use AAS, all that is necessary is identifying two equal angles in a triangle, then finding a third side adjacent to only one of the angles in each of the triangles such that the two sides are equal. This is enough to prove the two triangles are congruent.

What is the 180 triangle theorem? ›

Angle Sum Theorem Statement

Statement: The angle sum theorem states that the sum of all the interior angles of a triangle is 180 degrees.

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How do you prove the angle sum property of a quadrilateral? ›

When we draw a draw the diagonals to the quadrilateral, it forms two triangles. Both these triangles have an angle sum of 180°. Therefore, the total angle sum of the quadrilateral is 360°.

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What is the proof of quadrilateral sum conjecture? ›

Conjecture (Quadrilateral Sum ): The sum of the measures of the interior angles in any convex quadrilateral is 360 degrees. Proof: The sum of the measures of the interior angles of any quadrilateral can be found by breaking the quadrilateral into two triangles.

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What are the properties of all quadrilaterals? ›

They have four vertices. They have four sides. The sum of all interior angles is 360°. They have two diagonals.

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How do you prove congruence in quadrilaterals? ›

Generally we have to put side's interior angles of one quadrilateral in correspondence with sides and angles of another and to prove that all correspondence with sides and angles of another and to prove that all corresponding pairs of sides and angles are congruent.

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How do you prove special quadrilaterals? ›

First, show the quadrilateral is a parallelogram. Then, use one of the following: If a parallelogram contains a pair of consecutive sides that are congruent, then it is a rhombus (reverse of the definition). If either diagonal of a parallelogram bisects two angles of the parallelogram, then it is a rhombus.

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What is the angle sum property of a parallelogram proof? ›

A parallelogram is a flat 2d shape which has four angles. The opposite interior angles are equal. The angles on the same side of the transversal are supplementary, that means they add up to 180 degrees. Hence, the sum of the interior angles of a parallelogram is 360 degrees.

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What is the sum of opposite angles of a quadrilateral proof? ›

If opposite angles sum up to 180° 180 ° , it will be proved. Here is quadrilateral ABCD A B C D inscribed in circle O O . If we assume angle subtended at centre by minor arc BAD=2θ B A D = 2 θ , angle subtended by major arc BCD=360°−2θ B C D = 360 ° − 2 θ .

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