Question

in triangle xyz, ( mangle z>mangle x + mangle y ). which must be true about ( \triangle xyz )?
( mangle x + mangle z<90^{circ} )
( mangle y>90^{circ} )
( angle x ) and ( angle y ) are complementary
( mangle x + mangle y<90^{circ} )

Explanation:

Step1: Recall triangle angle sum

In any triangle, the sum of interior angles is \(180^\circ\), so \(m\angle X + m\angle Y + m\angle Z = 180^\circ\).

Step2: Use given inequality

Given \(m\angle Z > m\angle X + m\angle Y\). Let \(S = m\angle X + m\angle Y\), then \(m\angle Z > S\) and \(S + m\angle Z = 180^\circ\). Substituting \(m\angle Z > S\) into \(S + m\angle Z = 180^\circ\), we get \(S + S < 180^\circ\) (since \(m\angle Z > S\), so \(S + m\angle Z > S + S\)).

Step3: Solve for S

\(2S < 180^\circ\), divide both sides by 2: \(S = m\angle X + m\angle Y < 90^\circ\).

Answer:

\(m\angle X + m\angle Y < 90^\circ\) (the last option)