Question

Find the value of $y$

Answer 1

Hint: In order to find the value of y, we would be using the properties of quadrilateral and a straight line. The property of a quadrilateral states that the sum of the interior angles of a quadrilateral is equal to ${360}^{\circ }$. Then using the Linear angle property we can find the value of y.

Complete step by step answer:
We are given the figure of quadrilaterals. Let’s name all the corner sides of the quadrilateral is ABCD, and the respective figure obtained is:

Now, we have a quadrilateral named ABCD. And, from the properties of quadrilaterals we know that the sum of all the interior angles of a quadrilateral is equal to ${360}^{\circ }$.That can be numerically written as:
$\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C+\mathrm{\angle }D={360}^{\circ }$
Substituting all the values of A, B, C and D from the figure in the above formula, we get:
${85}^{\circ }+{115}^{\circ }+{97}^{\circ }+\mathrm{\angle }D={360}^{\circ }$
Solving the above equation to obtain the value of interior angle of D, and we get:
$\Rightarrow {297}^{\circ }+\mathrm{\angle }D={360}^{\circ }$
Subtracting both the sides by ${297}^{\circ }$ in the above equation:
$\Rightarrow {297}^{\circ }+\mathrm{\angle }D-{297}^{\circ }={360}^{\circ }-{297}^{\circ }$
$\Rightarrow \mathrm{\angle }D={360}^{\circ }-{297}^{\circ }$
$\Rightarrow \mathrm{\angle }D={63}^{\circ }$ …..(1)
And, we obtain that the value of the interior angle D is ${63}^{\circ }$.

Now, at the point D, using the linear angle property of a straight line, we can write the equation as:
$\Rightarrow y^{\circ }+\mathrm{\angle }D={180}^{\circ }$ ……(2)
As, we know that according to the linear angle property all the angles lying at a point sum to be ${180}^{\circ }$
Now, substituting the value of angle D from equation 1 into the equation 2, we get:
$\Rightarrow y^{\circ }+{63}^{\circ }={180}^{\circ }$
Subtracting both sides by ${63}^{\circ }$, we get:
$\Rightarrow y^{\circ }+{63}^{\circ }-{63}^{\circ }={180}^{\circ }-{63}^{\circ }$
$\therefore y^{\circ }={117}^{\circ }$

Therefore, the value of y is ${117}^{\circ }$.

Note:If ever given to find any value for another polygon then we can use the formula of $\left(n-2\right){180}^{\circ }$, where n is the number of sides of a polygon.We can cross check the answer by substituting the value and solving it further.