To solve for $x$, I will first analyze the image.

In the given diagram, we have two triangles. The top triangle has vertices labeled $X$, $Y$, and $Z$ with side $YZ = 12$ and the angle at vertex $X$ labeled as $6.4^\circ$. The bottom triangle has vertices labeled $E$, $F$, and $G$, with side $FG = 15$ and an angle at vertex $E$ labeled as $x$.

Since the triangles appear to be similar due to the proportionality of the sides (both triangles are isosceles) and the fact that they share a common angle configuration (equal vertex angles), we can use the property of similar triangles to find the unknown angle $x$.

Similarity and Proportion of Angles:

• $\triangle XYZ \sim \triangle EFG$
• The angles of similar triangles are proportional, so $\angle X = \angle E$.

Thus, $x = 6.4^\circ$ due to the angle similarity.

Final Answer:

• $x = 6.4^\circ$

Would you like further explanation or more details on the solution? Here are some related questions you might explore:

1. What are the properties of similar triangles?
2. How do we determine if two triangles are similar?
3. Can we use the law of sines or cosines to solve this problem?
4. How can side length ratios help in proving triangle similarity?
5. How do isosceles triangles affect angle relationships?

Tip: In similar triangles, corresponding angles are always equal, and the sides are proportional.