Question

in triangle xyz, m∠z > m∠x + m∠y. which must be true about △xyz?
○ m∠x + m∠z < 90°
○ m∠x + m∠y < 90°

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is \(m\angle X + m\angle Y+m\angle Z=180^{\circ}\), so \(m\angle X + m\angle Y=180^{\circ}-m\angle Z\).

Step2: Use the given inequality

We are given that \(m\angle Z>m\angle X + m\angle Y\). Substitute \(m\angle X + m\angle Y = 180^{\circ}-m\angle Z\) into the inequality: \(m\angle Z>180^{\circ}-m\angle Z\).

Step3: Solve the inequality for \(m\angle Z\)

Add \(m\angle Z\) to both sides: \(2m\angle Z>180^{\circ}\), then \(m\angle Z > 90^{\circ}\).
Since \(m\angle X + m\angle Y=180^{\circ}-m\angle Z\) and \(m\angle Z>90^{\circ}\), we have \(m\angle X + m\angle Y<90^{\circ}\).

Answer:

\(m\angle X + m\angle Y<90^{\circ}\)