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In the diagram, we see three angles at point \( G \):

1. \( \angle BGC = 83^\circ \) (given)
2. \( \angle CGD = 41^\circ \) (given)
3. \( \angle DGB = (2x - 8)^\circ \) (unknown)

Since all these angles form a full circle around point \( G \), the sum of the angles must equal \( 360^\circ \).

We can write the equation:

\[
(2x - 8)^\circ + 83^\circ + 41^\circ = 180^\circ
\]

Now, simplify this:

\[
2x - 8 + 83 + 41 = 180
\]
\[
2x + 116 - 8 = 180
\]
\[
2x + 108 = 180
\]
\[
2x = 180 - 108
\]
\[
2x = 72
\]
\[
x = \frac{72}{2} = 36
\]

Thus, the value of \( x \) is \( \boxed{36} \).

Would you like more details or explanations on any step? Here are five related questions for further practice:

1. How would you set up the equation if the sum of angles formed a different shape?
2. What if the given angles were on a straight line instead of a circle?
3. How do you verify the solution once you have the value of \( x \)?
4. How would the solution change if one of the angles was \( (3x - 5)^\circ \)?
5. How can you extend this method to find angles in a polygon?

**Tip:** Always check if the angles form a specific geometric shape, such as a circle or a straight line, to use the correct sum for the angles.

Look at the diagram: Solve for x based on the angle equation where three angles are given as 83°, 41°, and (2x - 8)° at point G. Which equation can be used to solve for x?

Solve for x using the angle sum property in a geometric diagram where angles 83°, 41°, and (2x - 8)° are given. Learn how to set up and solve the equation step-by-step. Suitable for Grades 7-9.

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