# Angle Sum Property Of A Quadrilateral

Quadrilateral is derived from the Latin word quadri-laterus, meaning four-lateral sides. Thus, a quadrilateral is an enclosed figure with four sides. It has four vertices, four angles, and four sides.

The given image shows various four sides figures, which are all quadrilaterals. Some figures are simple, like square and rectangle, while some are complex, like trapezium, rhombus, and parallelogram.

• Irregular Trapezium
• Isosceles trapezium
• Parallelogram
• Rhombus
• Rhomboid
• Square
• Rectangle
• Oblong
• Kite
• Tangential trapezoid
• Right kite
• Watt and Hjelmslev

• Crossed trapezoid
• Antiparallelogram
• Crossed rectangle
• Crossed square

## Angle sum property of a quadrilateral

The sum of all angles inside a quadrilateral is equal to 360 degrees. Consider a square with side ABCD, then the sum of angles A, B, C and D will be equal to 360 degrees.

## Proof

In the above image, we can see in quadrilateral ABCD,

• Angle ABC, angle BCD, angle CDA, and angle DAB are the internal angles.
• AC is a diagonal inside the quadrilateral.

We know that the sum of angles in a triangle is 180°.

Considering triangle ADC, we get ∠D + ∠DAC + ∠DCA = 180° (Sum of angles in a triangle)

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

Add both the above equations, we get,

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

∠D + (∠DAC + ∠BAC) + (∠BCA + ∠DCA) + ∠B = 360°

We can see that (∠DAC + ∠BAC) = ∠DAB and (∠BCA + ∠DCA) = ∠BCD

Replacing them we have, ∠D + ∠DAB + ∠BCD + ∠B = 360°

That is, ∠D + ∠A + ∠C + ∠B = 360°.

Hence, proved that the sum of angles inside a quadrilateral is equal to 360 degrees.

Note: - There is no relation between adjacent and opposite angles of a quadrilateral. The angle sum property is valid only for all the angles of a quadrilateral.

Example

Find the remaining angle if other angles are 90, 60 and 45 degrees.

Solution

We know the sum of all angles inside a quadrilateral is equal to 360 degrees.

Let the unknown angle be x.

Therefore, 90° + 45° + 60° + x = 360°

195° + x = 360°

x = 360° – 195°

x = 165°

Example

The angles of a quadrilateral are given by (3x + 2)°, (x – 3), (2x + 1)°, 2(2x + 5)°. What is the value of x and the measure of each angle?

Solution:

We know the angle sum property of a quadrilateral,

Therefore, we get (3x + 2)°+ (x – 3)° + (2x + 1)° + 2(2x + 5)°= 360°

⇒ 3x + 2 + x - 3 + 2x + 1 + 4x + 10 = 360°

⇒ 10x + 10 = 360

⇒ 10x = 360 – 10

⇒ 10x = 350

⇒ x = 350/10

⇒ x = 35

Therefore, (3x + 2) = 3 × 35 + 2 = 105 + 2 = 107°

(x – 3) = 35 – 3 = 32°

(2x + 1) = 2 × 35 + 1 = 70 + 1 = 71°

2(2x + 5) = 2(2 × 35 + 5) = 2(70 + 5) = 2 × 75 = 150°

Therefore, the angles of a quadrilateral are 32°, 71° 107° and 150°.

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