Why Do the Angles of a Quadrilateral Add Up to 360 Degrees?
Angle Sum Property of Quadrilateral
We have already learned that three noncollinear points make a triangle on joining. Now let us mark four non-collinear points such that no three of them are collinear and see what we get by joining them. We get a closed figure with four sides by joining four non-collinear points, such a figure is called a quadrilateral.
All around us we see quadrilateral shapes like floor, ceiling, window, blackboard, table, and many more.
On this page, we will learn about quadrilaterals, their different properties, and the derivation of angle sum property of quadrilaterals.
Quadrilateral Definition
The word ‘ quad’ means four and the word ‘lateral’ means sides i.e. A quadrilateral is a figure bounded by four line segments such that no three of them are parallel.
A quadrilateral has four sides, four vertices, and four angles.
Thus below figure ABCD is a quadrilateral that is bounded by four sides i.e AB, BC, CD, and AD. The four vertices are A, B, C, and D. ∠A, ∠B, ∠C, and ∠D are the four angles of the quadrilateral. And it is written as □ABCD and read as quadrilateral ABCD.
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A line segment drawn from one vertex to the opposite vertex is called the diagonal of the quadrilateral. From the below figure Segment AC and BD are the diagonals of the quadrilateral ABCD
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Terms Related to Quadrilateral
Opposite Sides: Two sides of a quadrilateral are opposite sides if the sides have no common vertex. In the above figure side AB and DC; Side AD and BC are the two pairs of opposite sides.
Opposite Angles: Two angles of a quadrilateral are opposite angles if they don’t have any common arm. In the above figure, ∠A and ∠C; ∠B and ∠D, are two pairs of opposite angles.
Adjacent Sides: Two Sides of a quadrilateral are said to be adjacent if the sides have a common vertex. In the above figure Side AB and BC; Side BC and CD; Side CD and DA; Side DA and AB are the four pairs of adjacent sides also called consecutive sides.
Adjacent Angles: Two angles of a quadrilateral is said to be adjacent angles if the angles have a common side or an arm. In the above figure ∠A and ∠B; ∠B and ∠C, ∠C and ∠D; ∠D and ∠A are the four pairs of adjacent angles also called consecutive angles.
Types of Quadrilateral
There are basically six types of quadrilaterals. They are as follows,
1. Parallelogram: A quadrilateral that has its opposite sides congruent and parallel to each other is a parallelogram. Its opposite angles are also congruent to each other.
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2. Rectangle: A quadrilateral that has its opposite sides equal and all the angles are at right angles(900) is called a rectangle
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3. Square: A quadrilateral that has all its four sides equal and opposite side are parallel, and all the angles at right angles(900) is called a square.
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4. Rhombus: A quadrilateral has all its sides equal and its diagonals bisect each other at 900 is called a rhombus.
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5. Trapezium: A quadrilateral that has only one pair of sides are parallel and the two sides are non-parallel is called a trapezium. The sides may not be equal to each other.
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6. Kite: A quadrilateral is a kite if it has two pairs of equal adjacent sides and unequal opposite sides.
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Quadrilateral Angles
As we know that a quadrilateral has four angles. The sum of the angles of the quadrilateral is 3600.
The sum of all the angles of the □ABCD ∠A +∠B + ∠C + ∠D = 360°
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In the case of square and rectangle, the measure of all the angles is 900.
Therefore we have ∠A = ∠B = ∠C = ∠D = 90°
Angle Sum Property of Quadrilateral Theorem
Angle sum Property of a Quadrilateral Theorem states that,
The sum of the measures of four angles of a quadrilateral is 3600
that is, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.
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Angle sum property of quadrilateral Proof
Proof: Let ABCD be a quadrilateral and AC be the diagonal of the quadrilateral ABCD
From the figure, we can see that the diagonal AC divides the quadrilateral ABCD in two triangles i.e ABCand ADC
Step 1:
By the angle sum property, the sum of angles in a triangle is 180°.
So consider ABC,
∠B + ∠BCA + ∠BAC= 180° ……(1)
Similarly in ADC
∠D + ∠DCA + ∠DAC = 180° …..(2)
Step 2:
Adding equation 1 and 2 we get
(∠B + ∠BCA + ∠BAC) + (∠D + ∠DCA + ∠DAC) = 180 + 180
(∠B + ∠BCA + ∠BAC) + (∠D + ∠DCA + ∠DAC) = 3600………..(3)
Step 3:
From the figure,
∠DAC + ∠BAC = ∠DAB
∠BCA + ∠DCA = ∠BCD.
Step 4:
Putting these values in equation 3
∠D + ∠BCD + ∠DAB + ∠B = 360°
That is, ∠D + ∠C + ∠A + ∠B = 360°.
∠A + ∠B + ∠C + ∠D = 360°.
Hence the angle sum property of the quadrilateral theorem proved.
The sum of the measures of four angles of a quadrilateral is 3600 Using the angle sum property of the quadrilateral theorem let us solve some examples.
Solved Examples
Example 1:
In the figure below x = 800, y = 1100, z =1200. Find w?
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Solution:
From Angle sum property of quadrilateral Proof
We have w + x + y + z = 3600
x = 800, y = 1100, z =1200
w + 80 + 110 + 120 = 360
w + 310 = 360
w = 360 - 310
w = 500
Example 2: Find the value of x
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Solution:
From Angle sum property of quadrilateral Proof
We have, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.
Given ∠ABC =1060, ∠BCD = x0, ∠CDA = 680, ∠DAB = 1260
So, 106 + x + 68 + 126 = 360
x + 300 = 360
x = 360 - 300
x = 600
Quiz
Using angle sum property of quadrilateral theorem solve the following
The three angles of a quadrilateral are 60˚, 70˚, 90˚. Find the fourth angle?
If the measure of two angles of a quadrilateral is 65˚ and 85˚ and the other two angles are equal, find the measure of each of the equal angles?
FAQs on Angle Sum Property of Quadrilaterals: Step-by-Step Guide
1. What is the Angle Sum Property of a Quadrilateral?
The Angle Sum Property of a quadrilateral states that the sum of the four interior angles of any quadrilateral is always 360 degrees. This is a fundamental property that holds true for all types of quadrilaterals, such as squares, rectangles, parallelograms, and trapeziums.
2. How can you prove that the sum of the angles in a quadrilateral is 360°?
The proof for the angle sum property of a quadrilateral is derived from the angle sum property of a triangle. The steps are:
- Consider a quadrilateral ABCD.
- Draw a diagonal, for instance, from vertex A to C. This diagonal divides the quadrilateral into two triangles: ΔABC and ΔADC.
- We know the sum of angles in any triangle is 180°.
- For ΔABC, the sum of its angles is 180°. For ΔADC, the sum of its angles is also 180°.
- The sum of the angles of the quadrilateral is the combined sum of the angles of these two triangles. Therefore, the total sum is 180° + 180° = 360°.
3. How do you use the angle sum property to find a missing angle in a quadrilateral?
To find a missing angle in a quadrilateral when the other three are known, you add the measures of the three given angles and subtract their total from 360°. For example, if a quadrilateral has angles measuring 80°, 100°, and 110°, and the fourth angle is 'x', the calculation would be:
80° + 100° + 110° + x = 360°
290° + x = 360°
x = 360° - 290°
x = 70°
4. Why is the sum of angles in a quadrilateral 360° and not 180°?
The sum of angles is 180° for a triangle, which is a three-sided polygon. A common misconception is to apply this to a quadrilateral. A quadrilateral is a four-sided polygon, which can be thought of as being composed of two triangles. Since each triangle has an angle sum of 180°, the total for the two triangles that make up the quadrilateral is 180° + 180° = 360°.
5. How is the angle sum property of a quadrilateral related to the angle sum property of a triangle?
The angle sum property of a quadrilateral is directly dependent on the angle sum property of a triangle. The proof itself relies on dividing the quadrilateral into two distinct triangles by drawing a diagonal. The total sum of the quadrilateral's angles is simply the sum of the angles of these two triangles, making the triangle's property the foundational concept for the quadrilateral's property.
6. Does the angle sum property of 360° apply to all types of quadrilaterals, including concave ones?
Yes, the angle sum property applies to both convex and concave quadrilaterals. A convex quadrilateral has all interior angles less than 180°. A concave quadrilateral has one interior angle that is a reflex angle (greater than 180°). Even in a concave quadrilateral, if you draw a diagonal that splits it into two triangles, the sum of the four interior angles will still correctly add up to 360°.
7. Is there an angle sum property for the exterior angles of a quadrilateral?
Yes, there is. For any convex quadrilateral, the sum of its four exterior angles is always 360°. This is a property that extends to all convex polygons, not just quadrilaterals. Each exterior angle is supplementary to its adjacent interior angle (they add up to 180°).
8. What is the real-world importance of the quadrilateral angle sum property?
The angle sum property is crucial in fields like architecture, engineering, and construction. It ensures the structural integrity and stability of four-sided frames, such as windows, doors, and building foundations. For example, architects rely on this property to ensure corners are perfectly square (90°) in rectangular designs, leading to a total of 360° and guaranteeing a stable, properly aligned structure.
