Find the value of $y$ if $x = 25^\circ$ and also find the value of $x$ if $y = 35^\circ$

There are two cases in this geometry problem. In the first case, the value of $x$ is given and it is $25^\circ$ but the value of $y$ is unknown. In the second case, the value of $x$ is unknown but the value of $y$ is given and it is $35^\circ$.

However, there is a geometrical relation between the literals $x$ and $y$ as displayed in the image. Let us first find the relation between theme and then the same relation used to find the values of variables in both cases.

Relation

$\overleftrightarrow{AB}$ is a straight line and its middle point is $O$. We know that, the angle of a straight line is always $180^\circ$ geometrically.

$\therefore \,\,\,\,\,\, \angle AOB = 180^\circ$.

The ray $\overrightarrow{OC}$ is started from point $O$. It makes $3x^\circ$ angle with $\overrightarrow{OB}$ line and also makes ${(2y+5)}^\circ$ angle with $\overrightarrow{OA}$ ray.

Therefore, $\angle COB = 3x^\circ$ and $\angle COA = {(2y+5)}^\circ$.

Geometrically, the angle $AOB$ is equal to the sum of the angles $COA$ and $COB$.

$\therefore \,\,\,\,\,\, \angle AOB = \angle COA + \angle COB$

$\implies \angle COA + \angle COB = \angle AOB$

$\implies {(2y+5)}^\circ + 3x^\circ = 180^\circ$

1. The mathematical meaning of $x^\circ$ in the term $3x^\circ$ is the value of $x$ is in degrees. Remember it and just write it as $x$ in the equation. Therefore, the product of number $3$ and $x$ gives an angle in degrees.
2. The meaning of the expression ${(2y+5)}^\circ$ is an angle in degrees. It is actually formed by the sum of the terms $2y$ and $5$. If the value of the expression ${(2y+5)}^\circ$ is in degrees, the values of the terms $2y$ and $5$ should be in degrees. Then only the sum of them will be in degrees. So, The term $5$ is written as $5^\circ$. Similarly, the term $2y$ is in degrees and it is possible if $2$ is a number and $y$ is an angle in degrees. Therefore, the expression ${(2y+5)}^\circ$ can be written as $2y+5^\circ$ in the equation.

$\implies 2y + 5^\circ + 3x = 180^\circ$

$\implies 3x+2y = 180^\circ -5^\circ$

$\therefore \,\,\,\,\,\, 3x+2y = 175^\circ$

Case: 1

It is given that the value of $x$ is $25^\circ$, substitute it in the equation and find the value of $y$ in degrees.

$3x+2y = 175^\circ$

$\implies 3(25^\circ) + 2y = 175^\circ$

$\implies 75^\circ + 2y = 175^\circ$

$\implies 2y = 175^\circ -75^\circ$

$\implies 2y = 100^\circ$

$\implies y = \dfrac{100^\circ}{2}$

$\therefore \,\,\,\,\,\, y = 50^\circ$

Therefore, the value of $y$ is $50$ degrees if the value of $x$ is $25^\circ$.

Case: 2

In this case, the value of $y = 35^\circ$. Substitute the value of $y$ in the equation to obtain the value of $x$ in degrees.

$3x+2y = 175^\circ$

$\implies 3x+2(35^\circ) = 175^\circ$

$\implies 3x+70^\circ = 175^\circ$

$\implies 3x = 175^\circ -70^\circ$

$\implies 3x = 105^\circ$

$\implies x = \dfrac{105^\circ}{3}$

$\therefore \,\,\,\,\,\, x = 35^\circ$

Therefore, the value of $x$ is $35^\circ$ if the value of $y$ is equal to $35^\circ$.

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