To find the statement [tex]$X \rightarrow Z$[/tex], we need to combine the given statements.
1. The first given statement is:
[tex]\[
X \rightarrow Y: \text{If the sum of the interior angles of a shape is } 180^{\circ}, \text{ then it is a triangle}.
\][/tex]
2. The second given statement is:
[tex]\[
Y \rightarrow Z: \text{If a shape is a triangle, then it has three sides}.
\][/tex]
We need to chain these implications together to find [tex]$X \rightarrow Z$[/tex].
- Starting from [tex]$X \rightarrow Y$[/tex], we know if the sum of the interior angles of a shape is [tex]$180^{\circ}$[/tex], we conclude it is a triangle.
- Then, using [tex]$Y \rightarrow Z$[/tex], if a shape is a triangle, we conclude it has three sides.
So, if a shape's interior angles sum to [tex]$180^{\circ}$[/tex], using the first statement [tex]$X \rightarrow Y$[/tex], it must be a triangle. Once we know it's a triangle, using the second statement [tex]$Y \rightarrow Z$[/tex], we conclude the shape has three sides.
Thus, the combined implication [tex]$X \rightarrow Z$[/tex] is:
[tex]\[
\text{If the sum of the interior angles of a shape is } 180^{\circ}, \text{ then it has three sides}.
\][/tex]
Therefore, the correct answer is:
C)$ \text{If the sum of the interior angles of a shape is } 180^{\circ}, \text{ then it has three sides}.
