We are given three angles:

• $\angle GEH = x + 40^\circ$
• $\angle DEG = x + 30^\circ$
• $\angle HEF = 2x + 10^\circ$

To solve for $x$, we need to know how these angles relate to each other. Typically, such problems involve angles in a straight line, forming a linear pair, or part of a triangle where the sum of angles is 180°.

Step-by-Step Solution:

Assuming these angles are on a straight line, such that: $\angle GEH + \angle DEG + \angle HEF = 180^\circ$

Substituting the expressions for the angles: $(x + 40) + (x + 30) + (2x + 10) = 180$

Simplify the equation: $x + 40 + x + 30 + 2x + 10 = 180$ $4x + 80 = 180$

Solve for $x$: $4x = 180 - 80$ $4x = 100$ $x = \frac{100}{4} = 25$

Finding the Value of $\angle GEH$:

Now that we know $x = 25$, substitute this value into the expression for $\angle GEH$: $\angle GEH = x + 40 = 25 + 40 = 65^\circ$

Thus, $\angle GEH = 65^\circ$.

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Would you like more details or explanations on any part of this solution? Here are 5 related questions to deepen your understanding:

1. How do we determine when angles form a straight line?
2. What is the general rule for the sum of angles in a triangle?
3. Can you explain the concept of a linear pair of angles?
4. How does substituting $x$ into the angle expressions help verify the solution?
5. What would change if these angles were part of a different geometric configuration?

Tip: Always check if the angles given form a straight line, a triangle, or another shape, as this determines how their sum behaves!