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.

$\overleftrightarrow{AB}$ 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 $COB$.

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

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

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

- 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.
- 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.

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

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

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

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 $25^\circ$.

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 $35^\circ$.

Latest Math Topics

Mar 21, 2023

Feb 25, 2023

Feb 17, 2023

Feb 10, 2023

Latest Math Problems

Mar 03, 2023

Mar 01, 2023

Feb 27, 2023

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved