Question 2 (5 points)

Considering the following statements, what is [tex]X \rightarrow Z[/tex]?

[tex]X \rightarrow Y[/tex]: If the sum of the interior angles of a shape is [tex]180^{\circ}[/tex], then it is a triangle.

[tex]Y \rightarrow Z[/tex]: If a shape is a triangle, then it has three sides.

A) [tex]X \rightarrow Z[/tex]: If a shape has three sides, then it is a triangle.

B) [tex]X \rightarrow Z[/tex]: If the sum of the interior angles of a shape isn't [tex]180^{\circ}[/tex], then it doesn't have three sides.

C) [tex]X \rightarrow Z[/tex]: If the sum of the interior angles of a shape is [tex]180^{\circ}[/tex], then it has three sides.

D) [tex]X \rightarrow Z[/tex]: If a shape has three sides, then the sum of the interior angles of the shape is [tex]180^{\circ}[/tex].



Answer :

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}.