Question

in the accompanying diagram, lines \\( \overleftrightarrow { a b } \\) and \\( \overleftrightarrow { c d } \\) intersect at point \\( e \\). if \\( m \angle c e b = 3 x - 14 \\) and \\( m \angle a e d = 31 \\), find the value of \\( x \\).

Explanation:

Step1: Use vertical angles property

Vertical angles are equal. $\angle CEB$ and $\angle AED$ are vertical angles. So, $m\angle CEB=m\angle AED$.

Step2: Substitute the given expressions

Given $m\angle CEB = 3x - 14$ and $m\angle AED = 31$. Then $3x-14 = 31$.

Step3: Solve for $x$

Add 14 to both sides of the equation: $3x-14 + 14=31 + 14$, which simplifies to $3x=45$.
Divide both sides by 3: $x=\frac{45}{3}$.

Answer:

$x = 15$