CLASS-8
GEOMETRY-ANGLES-PROBLEM & SOLUTION

There are some examples are given below for your better understanding –

Example.1) If an angle is 4/5 of its supplement, find the angles.

Ans.) Let the angle = x⁰, so its supplement = 180⁰ - x⁰

Given that  x⁰ =  4/5 (180⁰ - x⁰)

So,       5x⁰ =  4 X 180⁰ - 4x⁰

So,       9x⁰ = 4 X 180⁰

So,        x⁰ = 720⁰ / 9 = 80⁰                   (Ans.)




Example.2)  Find the values of x& y from the adjoining figure, where POQ is a straight line.

Ans.) here, OT stands on the line PQ. So, ∠POT + ∠QOT = 180⁰,

So, 3x + 22⁰ + 47⁰ = 180⁰

So, 3x = 180⁰ - (22⁰ + 47⁰) = 111⁰

So, x = 111⁰/3 = 37⁰

Being, angles at a point on one side of the line PQ, ∠POR + ∠ROS + ∠SOQ = 180⁰

 90⁰ + x + y + 18⁰ = 180⁰

Or,   90⁰ + 37⁰ + y + 18⁰ = 180⁰

Or,   y = 180⁰ - 145⁰ =  35⁰

 Hence, x = 37⁰ and y = 35⁰                (Ans.)




Example.3) Find the values of x & y from the adjoining figure when x – y = 10⁰

Ans.)    ∠COE = vertically opposite ∠DOF = 90⁰

So, sum of the angles at a point on one side of a straight line = 180⁰,

         ∠AOE + ∠COE + ∠BOC = 180⁰

=> x + 22⁰ + 90⁰ + y + 36⁰ = 180⁰

=>  x + y + 148⁰ =  180⁰

=>   x + y = 32⁰…………………….(1)

And, as per given condition we have,  x – y = 10⁰ …………………………..(2)

Now, we will add (1) & (2), and we find -

                    x + y  =  32⁰

                    x – y  =  10⁰

                 +    -     +

           ----------------------

                    2x  =  42⁰

             So,    x  =  21⁰

   Hence, if we put the value of x in equation (1), then we find –                        

          x + y  =  32⁰

or,     21⁰ + y = 32⁰

or,           y  =  32⁰ - 21⁰ =  11⁰

hence,       x  =  21⁰  &  y =  11⁰             (Ans.)



Example.4)  Find the value of x, y, & z from the adjoining figure, where  x : y : z  =  1 : 2 : 3

Ans.)  Let, x = a, y = 2a, and z = 3a

Then, 2z + 3y – 3x = 6a + 6a – 3a  =  9a,

5z = 15a,   3y = 6a

So, 2x + y = 2a + 2a  = 4a,  and  2x = 2a

So, sum of the angles at a point = 360⁰

So, 9a + 15a + 6a + 4a + 2a =  360⁰

Or,    36a  =  360⁰

Or,    a = 10⁰   or    x =  10⁰

Hence we can find also,  y =  20⁰, and  z = 30⁰

So,  x = 10⁰,  y =  20⁰, and  z = 30⁰                  (Ans.)