Question

24
in the accompanying diagram, \\( \overleftrightarrow { a f } \\) and \\( \overleftrightarrow { l g } \\) intersect at \\( r \\). \\( m \angle l r f = 7 x - 84 \\) and \\( m \angle a r g = 3 x \\). what is the value of \\( x \\)?

Explanation:

Step1: Identify vertical angles

Vertical angles are equal. $\angle LRF$ and $\angle ARG$ are vertical angles.

Step2: Set up the equation

Since $m\angle LRF = m\angle ARG$, we have $7x - 84=3x$.

Step3: Solve the equation

Subtract $3x$ from both sides: $7x-3x - 84=3x - 3x$.
$4x-84 = 0$.
Add $84$ to both sides: $4x-84 + 84=0 + 84$.
$4x=84$.
Divide both sides by $4$: $x=\frac{84}{4}=21$.

Answer:

$21$