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Question 1

  A quadrilateral has three acute angles, each measures 80°. What is the measure of the fourth angle?            

Solution:
Sum of the four angles of a quadrilateral = 360°

                 80° + 80° + 80° + 4th angle = 360°

... 4th angle = 360° - (80°+80°+80°)=360°-240°=120° 


Question 2

  In a quadrilateral ABCD, the measure of the three angles A, B and C of the quadrilateral are 110°, 70° and 80° respectively. Find the measure of the third angle. 

Solution:
The measure of A = 110°

The measure of B = 70°

The measure of C = 80°

The sum of the four angles of the quadrilateral ABCD = ∠A + ∠B + ∠C +∠D=360°.

∠A + ∠B + ∠C  = 110°+70°+80° = 260°

∠A + ∠B + ∠C +∠D = 360°

∠D = 360°-(∠A + ∠B + ∠C)

= 360°-260°

= 100°


Question 3

  In a quadrilateral ABCD, ∠ D is equal to 150° and A = B = C. Find ∠ A, ∠ B and ∠ C.

Solution:
Measure of ∠ D = 150°

Let
A= B = C = x°

Sum of the angles of the quadrilateral is 360°.


x° +x° +x° +150° = 360°

3x° +150° = 360°

3x° = 360° -150° = 210°

x = = 70°

∴ ∠ A = 70°, B = 70° and C = 70°.



Solution:
Given the ratio of the angles of a quadrilateral = 1:2:3:4

Therefore, let the angles of the quadrilateral be x, 2x, 3x and 4x.

The sum of the angles of a quadrilateral is 360°.


x+2x+3x+4x = 360°

10x = 360°

x = 36°

2x = 2 × 36° = 72°

3x = 3 × 36° = 108°

4x = 4 × 36° = 144°

The measures of the four angles are 36°, 72°, 108° and 144°.   


Question 5

  The In a quadrilateral
(i) which of them have their diagonals bisecting each other?
(ii) which of them have their diagonals perpendicular to each other?
(iii) which of them have equal diagonals ?
 
Solution:

Diagonals bisect each other in
a) parallelogram
b) rhombus
c) rectangle
d) Square
e) Kite

(ii) Diagonals are perpendicular in
a) rhombus
b) Square
c) Kite

(iii) Diagonals are equal to each other in
a) rectangle.
b) square



Question 6

  Adjacent sides of a rectangle are in the ratio 5: 12, if the perimeter of the rectangle is 34cm, find the  length of  the diagonal.          


Solution:
Given the adjacent sides of a rectangle are in the ratio 5:12.
Therefore let the sides be 5x and 12x.
Then 5x + 12x + 5x + 12x = 34

                                 34x = 34

                                    x = 1cm

Hence the sides are 12cm and 5cm.

5cm
                       12 cm
The length of the diagonal =
(52 +  122) ) (In a right angled triangle applying Pythagoras theorem)

                          = (25 +  144)

                          = 169 = 13cm.    

Therefore the length of the diagonal is 13cm.

Question 7

  The opposite angles of a parallelogram are (3x + 5)o and (61 - x)o. Find the measure of four angles.


Solution:
(3x + 5)  =  (61 - x) (Opposite angles of a parallelogram are equal)
 3x + x = 61 - 5
          4x = 56o

x=
            x = 14o

3x + 5  =  3(14) + 5  =  42 + 5  =  47o
 61 - x  =  61 – 14  =  47o

Angle adjacent to one of the above  angle = 180o – 47o
                                                               = 133o  (Sum of adjacent angles in a parallelogram is 180o)

Fourth angle = 133o (Opposite angles of a parallelogram are equal)

Therefore the four angles in a parallelogram are 47o, 133o, 47o and 133o

Question 8

  ABCD is a ||gm with ∠ A = 800. The internal bisectors of ∠ B and ∠ C meet at O. Find the measure of the three angles of Δ BCO.


Solution:
∠ C = ∠ A (Opposite angles of a ||gm are equal)

∠ C = 800 (Given ∠ C = 800)

∠ OCB = = = 400

                                

∠ B = 1800 - ∠ A    (Sum of interior angles on the same side of the transversal is 1800 )
      = 1800 - 80
      = 1000

∠ CBO = = = 5O0 ∠ BOC = 1800 – (∠ OBC + ∠ CBO)  (Angle sum of a Δ )

           = 1800 – (400 + 500)

           = 1800 - 900

           = 900

The Three angles of the triangle BCO namely ∠ OCB, ∠ CBO, ∠ BOC are 400, 500 and 900 respectively.

Question 9

  Find the measure of all four angles of a parallelogram whose consecutive angles are in the ratio 1 : 3.


Solution:
Given consecutive angles of a parallelogram are in the ratio 1:3
Therefore, the two consecutive angles be x and 3x.


x + 3x = 1800        (sum of the interior angles on the same side of the transversal is 1800)

      4x = 1800

        x = 450

Therefore the two consecutive angles are 450 and 3(450) = 1350.

Since the opposite angles of a parallelogram are equal. The measures of all four angles of a parallelogram are 450, 450, 1350 and 1350.

Question 10

  A diagonal and a side of a rhombus are of equal length. Find the measure of the angles of the rhombus.

 


Solution:

Let ABCD be the rhombus.

AB = BC = DC = DA   (sides of a rhombus are equal)

But AB = BD (Given)

AB = BC = CD = DA = BD

Since in Δ ABD all the sides are equal. Δ ABD is an equilateral Δ .

Similarly Δ BCD is also an equilateral.

               

     \ ∠ A = ∠ ABD = ∠ ADB = ∠ DBC = ∠ C = ∠ CDB = 600

     ∴ ∠ B = ∠ ABD + ∠ DDC = 600 + 600 = 1200

and ∠ D = ∠ ADB + ∠ CDB = 600 + 600 = 1200

The angles of the rhombus are 600, 1200, 600 and 1200.





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