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 properties of triangle angles

The sum of angles in a triangle is \(180^{\circ}\). In right - triangle \(XYZ\) with \(\angle Y = 90^{\circ}\), 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 measure of \(\angle X\) and \(\angle Z\)

From \(x + z=90^{\circ}\), since \(x>0\) and \(z > 0\), both \(\angle X\) and \(\angle Z\) 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 equal to \(90^{\circ}\).