Quadrilateral

Any closed figure in plane, bounded by four line segments is called a Quadrilateral. In the given quadrilateral ABCD :p1

i) Four vertices are A, \ B, \ C, \ and\ D

ii) Four line segments are AB,\ BC,\ CD,\ DA

iii) Four angles are \angle A,\ \angle B, \ \angle C \ and \ \angle D

iv) Two diagonals are AC \ and\  BD Let’s also learn the nomenclature related to quadrilaterals:

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1) Adjacent Sides: Two sides of a quadrilateral having a common vertex. Example AD \ and\ DC are adjacent sides with D as the common vertex. Similarly, AB \ and\ BC are adjacent sides with common vertex B.

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2) Opposite Sides: Two sides of a quadrilateral that doesn’t share a common vertex. Example: AB \ and\ DC are opposite side. Similarly, AD \ and\ BC are opposite sides. You will also see that they don’t share any common vertex.

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3) Adjacent Angles: Two angles of a quadrilateral having a common arm are called adjacent angles Example: In the above figure \angle A \ and\ \angle B are adjacent and they share AB as the common arm. Similarly, \angle B \ and\ \angle C are adjacent angles and they share BC as the common arm.

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4) Opposite Angles: Angles in a quadrilateral that doesn’t share any common arm are called opposite angles. Example: In the above figure, \angle A \ and\ \angle C are opposite angles. Similarly, \angle D \ and\ \angle B are opposite angles.

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Important Theorem:

Remember, in Polygons we read

Sum of the interior angles = (n-2) \times 180=(2n-4) \times 90^{\circ}

In a quadrilateral, n=4

Substituting in the above formula, we get that the sum of the internal angles is 360^{\circ}

Another way of proving this is as follows:p2

In \Delta ABC \colon \angle 2 + \angle 4 + \angle B = 180^{\circ} ...i)

Similarly, in \Delta ADC: \angle 1 + \angle 3 + \angle D = 180^{\circ} ...ii)

Adding i) and ii) we get

(\angle 1 + \angle 2) + (\angle 4 + \angle 3) + \angle D + \angle B   = 360^{\circ}   

Or \angle A+\angle B+\angle C+\angle D=360^{\circ}

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