Which statement proves that triangle $xyz$ is an isosceles right triangle?

• A. Triangle $xyz$ has a right angle and two sides of equal length.
• B. Triangle $xyz$ has two sides of different lengths and no right angle.
• C. Triangle $xyz$ has all three sides of equal length.
• D. Triangle $xyz$ has three angles of equal measure.

Answer: (A)

Verified step by step guidance

1. Recall the definition of an isosceles right triangle: it is a triangle that has one right angle (90 degrees) and two sides of equal length.
2. Identify that having a right angle means one angle measures exactly 90 degrees, which is a key characteristic of right triangles.
3. Recognize that having two sides of equal length means the triangle is isosceles, which implies two angles opposite those sides are also equal.
4. Combine these two facts: a triangle with a right angle and two equal sides must be an isosceles right triangle.
5. Therefore, the statement that proves triangle \(x y z\) is an isosceles right triangle is the one that says it has a right angle and two sides of equal length.