This diagram shows two pairs of vertical angles. The angles labeled as $2x^\circ$ and $(78 - x)^\circ$ are vertical angles, which means they are equal. To solve for $x$, we set up the equation:

$2x = 78 - x$

Solving the equation:

1. Add $x$ to both sides to eliminate the negative $x$ on the right: $2x + x = 78$ $3x = 78$

2. Divide both sides by 3 to solve for $x$: $x = \frac{78}{3} = 26$

Now that we have $x = 26$, we can substitute this value back into the expressions for the labeled angles:

• The first angle is $2x = 2(26) = 52^\circ$.
• The second angle is $(78 - x) = 78 - 26 = 52^\circ$.

Thus, the two labeled angles both measure $52^\circ$.

Final answers:

• $x = 26$
• First angle: $52^\circ$
• Second angle: $52^\circ$

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. What are vertical angles, and why are they always equal?
2. How would the solution change if one angle were labeled differently?
3. Can you determine the measure of the other angles in the diagram?
4. What are supplementary angles, and how could they apply in this scenario?
5. How can you verify the solution using geometric properties?

Tip: Vertical angles are always equal because they are formed by the intersection of two straight lines, making them congruent!