Question

ethan created three triangles: triangle x, triangle y, and triangle z. if △y ≅ △x and △y ≅ △z. which must also be true?
triangles x, y, and z share one or more vertices.
the triangles are right triangles.
the triangles are equilateral.
triangles x and z are congruent.

Explanation:

Step1: Use the transitive property of congruence

If \(\triangle X\cong\triangle Y\) and \(\triangle Y\cong\triangle Z\), by the transitive property of congruence (\(a = b\) and \(b = c\) implies \(a = c\) for congruent triangles), then \(\triangle X\cong\triangle Z\).

Step2: Analyze other options

• For the option "Triangles \(X\), \(Y\), and \(Z\) share one or more vertices": Congruence does not depend on sharing vertices.
• For the option "The triangles are right - triangles": The given information \(\triangle X\cong\triangle Y\) and \(\triangle Y\cong\triangle Z\) does not imply anything about the triangles being right - triangles.
• For the option "The triangles are equilateral": The given congruence relations \(\triangle X\cong\triangle Y\) and \(\triangle Y\cong\triangle Z\) do not imply that the triangles are equilateral.

Answer:

Triangles \(X\) and \(Z\) are congruent.