Question

In the figure $l\parallel m$ and p is a transversal, find the value of x, giving necessary reasons.

Answer 1

Hint: As the lines given are parallel and a transversal is also given, we can use the property of corresponding angles from which we can form an equation to solve for x.

Complete step-by-step answer:
In the question, we have been given a figure with certain conditions and we are asked to find the value of x from that. The figure is given as below,

We can see that that there are three lines in the given figure, out of which two lines, l and m are parallel to each other and p is a transversal with some points marked on it as A, B, C, D, E, F, G and H. Since we have been given that l is parallel to m and p is the transversal, we can apply the properties of parallel lines, which states that whenever parallel lines are cut by a transversal, the corresponding angles are equal. So, in this figure, we can say that $\angle AGB$ is equal to $\angle GHE$. We can see that the values of these angles are given as, ${{\left( 5x \right)}^{\circ }}$ and ${{\left( 2x+24 \right)}^{\circ }}$ respectively. So, we can say that they are equal and form an equation as follows,
$5x=\left( 2x+24 \right)$
Subtracting 2x from both sides of the equation, we get,
$\begin{align}
  & 5x-2x=2x+24-2x \\
 & \Rightarrow 3x=24 \\
 & \Rightarrow x=\dfrac{24}{3} \\
 & \Rightarrow x=8 \\
\end{align}$
Hence, we get the value of x as 8.

Note: We can also solve this question by using the properties of vertically opposite angles and alternate angles. We can first say that$\angle AGB$ is equal to $\angle CGH$as they are vertically opposite angles. Since $\angle AGB$ is equal to 5x, $\angle CGH$ would also be equal to ${{\left( 5x \right)}^{\circ }}$. Then we apply the property of alternate angles, that is $\angle CGH$equal to $\angle GHE$. As we know that $\angle GHE$ is equal to ${{\left( 2x+24 \right)}^{\circ }}$, we can form the equation accordingly as, $5x=\left( 2x+24 \right)$.