Chapter - 15 Quadrilaterals | Mathematics (Maths) Class 8 PDF Download

 Page 1


Q u e s t i o n : 1
Fill in the blanks:
i
A quadrilateral has ......... sides.
ii
A quadrilateral has ......... angles.
iii
A quadrilateral has ......... vertices, no three of which are .........
iv
A quadrilateral has ......... diagonals.
v
A diagonal of a quadrilateral is a line segment that joins two ......... vertices of the quadrilateral.
vi
The sum of the angles of a quadrilateral is.........
S o l u t i o n :
i
4
ii
4
iii
4, co-linear
iv
2
v
opposite
vi
360°
Q u e s t i o n : 2
In the adjoining figure, ABCD is a quadrilateral.
i
How many pairs of adjacent sides are there? Name them.
ii
How many pairs of opposite sides are there? Name them.
iii
How many pairs of adjacent angles are there? Name them.
iv
How many pairs of opposite angles are there? Name them.
v
How many diagonals are there? Name them.
S o l u t i o n :
i
There are four pairs of adjacent sides, namely (AB,BC),
BC, CD,
CD, DA and (DA,AB).
ii
There are two pairs of opposite sides, namely (AB,DC) and (AD,BC).
iii
There are four pairs of adjacent angles, namely ( ?A, ?B), ( ?B, ?C), ( ?C, ?D) and ( ?D, ?A)
.
iv
There are two pairs of opposite angles, namely ( ?A, ?C) and ( ?B, ?D)
.
v
There are two diagonals, namely AC and BD.
Q u e s t i o n : 3
Prove that the sum of the angles of a quadrilateral is 360°.
S o l u t i o n :
      
Let ABCD be a quadrilateral.
Join A and C.
Now, we know that the sum of the angles of a triangle is 180°.
For ? ABC: ?2 + ?4 + ?B = 180
o
            . . . 1
For ? ADC: ?1 + ?3 + ?D = 180
o
         . . . 2
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Page 2


Q u e s t i o n : 1
Fill in the blanks:
i
A quadrilateral has ......... sides.
ii
A quadrilateral has ......... angles.
iii
A quadrilateral has ......... vertices, no three of which are .........
iv
A quadrilateral has ......... diagonals.
v
A diagonal of a quadrilateral is a line segment that joins two ......... vertices of the quadrilateral.
vi
The sum of the angles of a quadrilateral is.........
S o l u t i o n :
i
4
ii
4
iii
4, co-linear
iv
2
v
opposite
vi
360°
Q u e s t i o n : 2
In the adjoining figure, ABCD is a quadrilateral.
i
How many pairs of adjacent sides are there? Name them.
ii
How many pairs of opposite sides are there? Name them.
iii
How many pairs of adjacent angles are there? Name them.
iv
How many pairs of opposite angles are there? Name them.
v
How many diagonals are there? Name them.
S o l u t i o n :
i
There are four pairs of adjacent sides, namely (AB,BC),
BC, CD,
CD, DA and (DA,AB).
ii
There are two pairs of opposite sides, namely (AB,DC) and (AD,BC).
iii
There are four pairs of adjacent angles, namely ( ?A, ?B), ( ?B, ?C), ( ?C, ?D) and ( ?D, ?A)
.
iv
There are two pairs of opposite angles, namely ( ?A, ?C) and ( ?B, ?D)
.
v
There are two diagonals, namely AC and BD.
Q u e s t i o n : 3
Prove that the sum of the angles of a quadrilateral is 360°.
S o l u t i o n :
      
Let ABCD be a quadrilateral.
Join A and C.
Now, we know that the sum of the angles of a triangle is 180°.
For ? ABC: ?2 + ?4 + ?B = 180
o
            . . . 1
For ? ADC: ?1 + ?3 + ?D = 180
o
         . . . 2
( )
( )
Adding 1
and 2
:
( ?1 + ?2 + ?3 + ?4)+ ?B + ?D = 360
o
or ?A + ?B + ?C + ?D = 360
o
Hence, the sum of all the angles of a quadrilateral is 360°.
Q u e s t i o n : 4
The three angles of a quadrilateral are 76°, 54° and 108°. Find the measure of the fourth angle.
S o l u t i o n :
Sum of all the four angles of a quadrilateral is 360°.
Let the unknown angle be x°. 76 +54 +108 +x = 360238 +x = 360x = 122
The fourth angle measures 122°.
Q u e s t i o n : 5
The angles of a quadrilateral are in the ratio 3 : 5 : 7 : 9. Find the measure of each of these angles.
S o l u t i o n :
Let the measures of the angles of the given quadrilateral be 3x °, 5x °, 7x ° and 9x °. Sum of all the angles of a quadrilateral is 360
o
. ? 3x +5x +7x +9x = 36024x = 360x = 15
Angles measure: 3 ×15 ° = 45
°
5 ×15 ° = 75
°
7 ×15 ° = 105
°
9 ×15 ° = 135
°
Q u e s t i o n : 6
A quadrilateral has three acute angles, each measuring 75°. Find the measure of the fourth angle.
S o l u t i o n :
Sum of the four angles of a quadrilateral is 360°.
If the unknown angle is x
°, then:
75 +75 +75 +x = 360x = 360 -225 = 135
The fourth angle measures 135°.
Q u e s t i o n : 7
Three angles of a quadrilateral are equal and the measure of the fourth angle is 120°. Find the measure of each of the equal angles.
S o l u t i o n :
Let the three angles measure x°
 each.
Sum of all the angles of a quadrilateral is 360°.
? x +x +x +120 = 3603x +120 = 3603x = 240x =
240
3
= 80
Each of the equal angles measure 80°.
Q u e s t i o n : 8
Two angles of a quadrilateral measure 85° and 75° respectively. The other two angles are equal. Find the measure of each of these equal angles.
S o l u t i o n :
Let the two unknown angles measure 
x° each.
Sum of the angles of a quadrilateral is 360°.
 
? 85 +75 +x +x = 360160 +2x = 3602x = 360 -160 = 200x = 100
Each of the equal angle measures 100°.
Q u e s t i o n : 9
In the adjacent figure, the bisectors of ?A and ?B meet in a point P. If ?C = 100° and ?D = 60°, find the measure of ?APB.
S o l u t i o n :
Sum of the angles of a quadrilateral is 360°.
? ?A + ?B +60
o
+100
o
= 360° ?A + ?B = 360 -100 -60 = 200°or
1
2
( ?A + ?B) = 100°        . . . 1 Sum of the angles of a triangle is 180°. In ? APB: 
1
2
( ?A + ?B)+ ?P = 180°                 because
? ?APB = 80°
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