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Understanding the Quadrilaterals Angle Sum Property

CEO Khai Intela
Introduction Have you ever wondered about the fascinating properties of quadrilaterals? In this article, we will explore one such property - the angle sum property of quadrilaterals. Before we delve into the details, let's review...

Introduction

Have you ever wondered about the fascinating properties of quadrilaterals? In this article, we will explore one such property - the angle sum property of quadrilaterals. Before we delve into the details, let's review some basics. An angle is formed when two line segments meet at a point, and it is measured in degrees (°). A quadrilateral, on the other hand, is a four-sided polygon with non-curved sides. It can have equal or unequal sides, but what's interesting is that the sum of all the angles inside a quadrilateral is always 360°! Let's discover more about this intriguing property.

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. This means that no matter what type of quadrilateral you have, the total sum of its angles will always be 360°.

Quadrilateral: Angle Sum Property Image Source: saigonintela.vn

Proof:

To understand why the angle sum property holds true, let's consider a quadrilateral ABCD. Here are the steps to prove the property:

  • Identify the internal angles of the quadrilateral: ∠ABC, ∠BCD, ∠CDA, and ∠DAB.
  • Draw a diagonal AC, dividing the quadrilateral into two triangles, ∆ABC and ∆ADC.
  • We know that the sum of angles in a triangle is 180°.
  • Applying this knowledge to triangle ADC, we have: ∠D + ∠DAC + ∠DCA = 180°.
  • Similarly, for triangle ABC, we have: ∠B + ∠BAC + ∠BCA = 180°.
  • Adding both equations together, we get: (∠D + ∠DAC + ∠DCA) + (∠B + ∠BAC + ∠BCA) = 180° + 180°.
  • Simplifying further, we obtain: ∠D + (∠DAC + ∠BAC) + (∠BCA + ∠DCA) + ∠B = 360°.
  • Replacing (∠DAC + ∠BAC) with ∠DAB and (∠BCA + ∠DCA) with ∠BCD, we have: ∠D + ∠DAB + ∠BCD + ∠B = 360°.
  • And finally: ∠D + ∠A + ∠C + ∠B = 360°.

Hence, the sum of angles of a quadrilateral is 360°, which is known as the angle sum property of quadrilaterals.

Quadrilateral Angles

A quadrilateral has a total of 4 angles, and the sum of its interior angles is always 360 degrees. By knowing the measurements of 3 angles or 2 angles or 1 angle and 4 lengths of the quadrilateral, we can find the values of the remaining angles. Let's take a look at a trapezoid, which is a type of quadrilateral, to better understand:

Quadrilateral Angles Image Source: saigonintela.vn

In the case of a square and a rectangle, all the angles are 90 degrees. Therefore, ∠A = ∠B = ∠C = ∠D = 90°. However, in general, a quadrilateral can have sides of different lengths and angles of different measures.

Do the Opposite sides in a Quadrilateral equal 180°?

Contrary to what you might think, there is no direct relationship between the opposite sides and the angle measures of a quadrilateral. For example, in a scalene trapezium, the side lengths are different, and the opposite angles are not 180 degrees. However, some special types of quadrilaterals, such as squares, isosceles trapeziums, rectangles, kites, rhombuses, and parallelograms, have supplementary opposite angles. This means that the sum of the angles is 180 degrees. Please note that it is not necessary for the opposite angles in a quadrilateral to be 180 degrees.

Types of Quadrilaterals

Quadrilaterals come in various types. Let's explore five of them:

  1. Parallelogram: This type has opposite sides that are equal and parallel to each other.
  2. Rectangle: All the angles in a rectangle are 90 degrees, and opposite sides are equal.
  3. Square: A square is characterized by all four sides being equal in length, and all angles measuring 90 degrees.
  4. Rhombus: A rhombus is a parallelogram with equal sides, and its diagonals bisect each other at 90 degrees.
  5. Trapezium: A trapezium has only one pair of parallel sides, and the remaining sides need not be equal.

Example

Let's apply our knowledge of quadrilateral angles to solve an example problem:

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

Solution: According to the angle sum property, the sum of all the interior angles of a quadrilateral is 360°.

Let the unknown angle be x. So, 90° + 45° + 60° + x = 360°. Simplifying further, we get: 195° + x = 360°. By transposing, we have: x = 360° - 195°. Calculating, we find: x = 165°.

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

Now that you understand the angle sum property of quadrilaterals and have gained insights into various types of quadrilaterals, you can appreciate the intricacies of these geometric shapes. Remember, the angle sum property is just one of the many intriguing aspects that make quadrilaterals so fascinating to study.

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