Page 220 - Start Up Mathematics_8 (Non CCE)

Quadrilaterals

A four-sided polygon is called a quadrilateral (Fig. 14.16). It has: B
– four sides (AB, BC, CD, DA) A
– four vertices (A, B, C, D)
– four angles (–DAB, –ABC, –BCD, –CDA)
– two diagonals (AC and BD) D C

The sides having a common vertex are adjacent sides. For example, Fig. 14.16
sides AB and BC, BC and CD, CD and DA, DA and AB
The sides with no common vertex are opposite sides. For example, Extension
sides AB and DC, AD and BC
A pair of interior angles with a common arm are adjacent angles. For A quadrilateral has
example, –DAB and –ABC, –ABC and –BCD, –BCD and –CDA, infinite number of
interior and exterior
–CDA and –DAB points.
A pair of interior angles with no common arm are opposite angles. For
example, –DAB and –BCD, –ABC and –CDA
All points lying inside the quadrilateral are interior points of the A X Y B T
quadrilateral (Fig. 14.17). For example, points X, Y, L, M K U

All points lying outside the quadrilateral are exterior points of the D L M C
quadrilateral (Fig. 14.17). For example, points T, K, V, U V
Fig. 14.17
The interior of a quadrilateral along with its boundary is the quadrilateral
region. N R

Convex and concave quadrilaterals G
A quadrilateral is convex, if its each interior angle is S Q
less than 180°. In a convex quadrilateral, every line
segment connecting any two points lie completely
within the quadrilateral (Fig. 14.18). I P
A quadrilateral which is not convex is concave K Fig. 14.18 Fig. 14.19
quadrilateral (Fig. 14.19).

Angle-sum property of a quadrilateral Q

The sum of all the interior angles of a quadrilateral is 360º or P 1
4 right angles. 2

Consider the quadrilateral PQRS (Fig. 14.20).
Draw a diagonal PR. It divides the quadrilateral PQRS into two 4 3
triangles, D PQR and D PSR. S R
In D PQR, –1 + –Q + –3 = 180° (Angle sum property of a triangle) Fig. 14.20
In D PSR, –2 + –S + –4 = 180° (Angle sum property of a triangle)

On adding, we get:
–1 + –Q + –3 + –2 + –S + –4 = 180° + 180°

(–1 + –2) + –Q + (–3 + –4) + –S = 360°
–1 + –2 = –P, –3 + –4 = –R

So, –P + –Q + –R + –S = 360° Or, –SPQ + –PQR + –QRS + –RSP = 360°

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