Page 181 - ICSE Math 8

A pair of interior angles with a common arm are called adjacent angles. For example, –DAB and –ABC, –ABC
and –BCD, –BCD and –CDA, –CDA and –DAB are adjacent angles.
A pair of interior angles with no common arm are called opposite angles. A B T
For example, –DAB and –BCD, –ABC and –CDA are opposite angles. K X Y
All points lying inside the quadrilateral are called interior points of L M U
the quadrilateral (Fig. 16.12). For example, points X, Y, L, M are all D V C
interior points. Fig. 16.12
All points lying outside the quadrilateral are called exterior points of the quadrilateral (Fig. 16.12). For example,
points T, K, V, U are all exterior points.
The interior of a quadrilateral along with its boundary is known as the quadrilateral region.

Convex and concave quadrilaterals
N R
A quadrilateral is convex, if its each interior angle is
less than 180°. In a convex quadrilateral, every line G
segment connecting any two points lie completely S Q
within the quadrilateral (Fig. 16.13).
A quadrilateral which is not convex is concave
quadrilateral (Fig. 16.14). K I P Fig. 16.14
Fig. 16.13
Angle-sum Property of a Quadrilateral

To prove: The sum of all the interior angles of a quadrilateral is 360º or Q
4 right angles. P 1
Proof: Consider the quadrilateral PQRS (Fig. 16.15). 2

Construction: Draw a diagonal PR. It divides the quadrilateral PQRS into two 4 3
triangles, D PQR and D PSR. S R
Fig. 16.15
Proof: In D PQR, –1 + –Q + –3 = 180° (Angle sum property of a triangle)
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° z B
60°
Example 1: Look at the figure given alongside and find the value of y
unknowns. Also find x + y + z + w. 80° C

Solution: Since x, y, z, w are the exterior angles of the quadrilateral
ABCD: x + y + z + w = 360° A 120°
w x
In quadrilateral ABCD, D
–A + –B + –C + –D = 360° (Sum of the four angles of a quadrilateral is 360°)
–A + 60° + 80° + 120° = 360° ⇒ –A + 260° = 360° ⇒ –A = 360° – 260° = 100°

Now, w = 180° – –A = 180° – 100° = 80°; z = 180° – –B = 180° – 60° = 120°
y = 180° – –C = 180° – 80° = 100°; x = 180° – –D = 180° – 120° = 60°
Therefore, x + y + z + w = 60° + 100° + 120° + 80° = 360°

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