Question

triangles and their side lengths
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 triangle angle sum

In any triangle, the sum of interior angles is \(180^\circ\). For right triangle \(XYZ\) with right angle at \(Y\), \(\angle Y = 90^\circ\).

Step2: Calculate sum of \(\angle X\) and \(\angle Z\)

Using the angle - sum property: \(\angle X+\angle Y+\angle Z = 180^\circ\). Substitute \(\angle Y = 90^\circ\), we get \(\angle X+\angle Z+90^\circ=180^\circ\). Then \(\angle X+\angle Z=180^\circ - 90^\circ=90^\circ\). So the sum of \(\angle X\) and \(\angle Z\) must be equal to \(90^\circ\) (i.e., the sum of angles \(X\) and \(Z\) must be equal to \(90^\circ\), so the blank for "The sum of angles \(X\) and \(Z\) must be" is "equal to \(90^\circ\)"). Also, since \(\angle X+\angle Z = 90^\circ\), \(\angle X\) and \(\angle Z\) are acute angles (less than \(90^\circ\)), so "Angle \(X\) must be" an acute angle (less than \(90^\circ\)) and "Angle \(Z\) must be" an acute angle (less than \(90^\circ\)).

Answer:

• The sum of angles \(X\) and \(Z\) must be: equal to \(90^\circ\)
• Angle \(X\) must be: an acute angle (less than \(90^\circ\))
• Angle \(Z\) must be: an acute angle (less than \(90^\circ\))