Question

determining angle relationships in right triangles
triangle xyz is a right triangle with the right angle at vertex y
angle x must be
angle z must be
the sum of angles x and z must be 90°

Explanation:

Step1: Recall the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). In \(\triangle XYZ\), \(\angle Y = 90^{\circ}\). Let \(\angle X=x\) and \(\angle Z = z\). Then \(x + y+z=180^{\circ}\), substituting \(y = 90^{\circ}\), we get \(x + z+90^{\circ}=180^{\circ}\).

Step2: Analyze the nature of non - right angles in a right - triangle

From \(x + z=180^{\circ}-90^{\circ}=90^{\circ}\). Also, since \(x>0\) and \(z > 0\), and \(x + z=90^{\circ}\), both \(x\) (angle \(X\)) and \(z\) (angle \(Z\)) are less than \(90^{\circ}\). Angles less than \(90^{\circ}\) are acute angles.

Answer:

Angle \(X\) must be acute.
Angle \(Z\) must be acute.
The sum of angles \(X\) and \(Z\) must be \(90^{\circ}\).