Math Doubts

Find the value of x if interior angles are x°, 3y° and 5y° and exterior angle of a triangle is 7y°

triangle with interior and exterior angles

$\Delta ABC$ is triangle and its interior angles are $x^\circ$, $3y^\circ$ and $5y^\circ$. The side $\overrightarrow{AB}$ of the triangle is extended through a point $D$. Thus, the triangle $ABC$ has an exterior angle $CBD$ and it is $7y^\circ$.

In this geometry problem, the literals $x$ and $y$ are used to represent angles but they are symbolically written as $x^\circ$ and $y^\circ$. The meaning of representing $x$ and $y$ as $x^\circ$ and $y^\circ$ is the values of $x$ and $y$ are in degrees, which means the literal numbers $x$ and $y$ represent angles in degree measuring system.

Now, let us find the value of $x$ by using the geometrical relation between $y$ and $x$.

Step: 1

Forget about the triangle $ABC$ but only consider the intersecting lines $\overrightarrow{AD}$ and $\overline{BC}$. Geometrically, the line segment $\overline{BC}$ divides the angle $ABD$ into two parts and the angles are $5y$ and $7y$ in degrees.

straight angle

Therefore $\angle CBA = 5y$ and $\angle CBD = 7y$ in degrees.

The line $\overrightarrow{AD}$ is a straight ray. So, the angle of the line $\overrightarrow{AD}$ is $\angle ABD$ and it is a straight angle. Therefore, the angle $ABD$ is $180^\circ$ geometrically.

Geometrically, the sum of the angles $\angle CBA$ and $\angle CBD$ is equal to $\angle ABD$.

$\angle CBD + \angle CBA = \angle ABD$

$\implies 7y + 5y = 180^\circ$

$\implies 12y = 180^\circ$

$\implies y = \dfrac{180^\circ}{12}$

$\implies \require{cancel} y = \dfrac{\cancel{180^\circ}}{\cancel{12}}$

$\therefore \,\,\,\,\,\, y = 15^\circ$

Therefore, the value of $y$ is $15^\circ$.

Step: 2

The value of literal $y$ in degrees is calculated in previous step and it is $15^\circ$. Now, consider only $\Delta ABC$. Geometrically, it is proved that the sum of three angles of a triangle is $180^\circ$.

triangle with unknown angles

It can be expressed in mathematical form as follows.

$\angle CBA + \angle BAC + \angle ACB = 180^\circ$

Substitute the values of all three angles of triangle $ABC$ in algebraic form.

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

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

$\implies x + 8y = 180^\circ$

$\implies x = 180^\circ -8y$

$\implies x = 180^\circ -8 (15^\circ)$

$\implies x = 180^\circ -120^\circ$

$\therefore \,\,\,\,\,\, x = 60^\circ$

Therefore, the value of $x$ is $60^\circ$ and it is the required solution for this geometric problem.

Latest Math Topics
Latest Math Problems
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more