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Question: There are four angles formed at a point say, w, x, y, z. If w = x = y and z = 2x then find the value...

There are four angles formed at a point say, w, x, y, z. If w = x = y and z = 2x then find the value of w, x, y, z.
A.w=x=y=72∘    z=144∘w=x=y=72{}^\circ \,\,\,\,z=144{}^\circ
B.w=x=y=144∘    z=72∘w=x=y=144{}^\circ \,\,\,\,z=72{}^\circ
C.w=x=y=60∘    z=120∘w=x=y=60{}^\circ \,\,\,\,z=120{}^\circ
D.w=x=y=90∘    z=180∘w=x=y=90{}^\circ \,\,\,\,z=180{}^\circ

Explanation

Solution

Hint: Use the concept “the sum of all the angles at a point is always equal to 360∘360{}^\circ ” and convert the equation in terms of ‘x’ by using the given data and solve to get the value of ‘x’. Then use the value of ‘x’ to get the final answer.

Complete step-by-step answer:
To solve the above question we will write the given data first, therefore,
w = x = y and z = 2x ………………………………………………….. (1)
Now to proceed further in the solution we should know the concept given below,
Concept: The measure of angle at any point is 360∘360{}^\circ and therefore the sum of all the angles at a point is always equal to 360∘360{}^\circ .
This concept can easily be understood from the diagram given below,

If we refer to the above concept then we can say that the sum of w, x, y, and z is also equal to 360∘360{}^\circ and therefore we can write,
⇒w+x+y+z=360∘\Rightarrow w+x+y+z=360{}^\circ
If we refer equation (1) then we know that w = x = y and therefore we can replace w and y by x, therefore we will get,
⇒x+x+x+z=360∘\Rightarrow x+x+x+z=360{}^\circ
Also if we put the value of z from equation (1) in the above equation we will get,
⇒x+x+x+2x=360∘\Rightarrow x+x+x+2x=360{}^\circ
If we perform the addition operation in the above equation we will get,
⇒5x=360∘\Rightarrow 5x=360{}^\circ
If we shift 5 on the right hand side of the equation we will get,
⇒x=360∘5\Rightarrow x=\dfrac{360{}^\circ }{5}
If we divide 360∘360{}^\circ by 5 in the above equation we will get,
⇒x=72∘\Rightarrow x=72{}^\circ
If we put this value in the equation (1) we will get,
w = 72∘72{}^\circ = y And z = 2×72∘2\times 72{}^\circ
Therefore we can write,
w = x = y = 72∘72{}^\circ And z = 144∘144{}^\circ
Therefore the value of w, x, y and z is given by w=x=y=72∘    z=144∘w=x=y=72{}^\circ \,\,\,\,z=144{}^\circ .
Therefore the correct answer is option (a).

Note: In the equation w+x+y+z=360∘w+x+y+z=360{}^\circ we substitute one value for w, x, and y but we have to put this value as ‘x’ as the value of z is also in terms of ‘x’ and therefore it will be easy to form equation in one variable and solve for it.