Question

determining angle measure relationships
in triangle rst, ( mangle r>mangle s + mangle t ). which must be true of triangle rst? choose four correct answers.
( mangle r>mangle t )
( mangle s + mangle t<90 )
( mangle r>mangle s )
( mangle s = mangle t )
( mangle s>mangle t )
( mangle r>90^{circ} )

Explanation:

Step1: Use the triangle angle - sum property

In any triangle \(ABC\), \(m\angle A + m\angle B+m\angle C = 180^{\circ}\). For \(\triangle RST\), \(m\angle R+m\angle S + m\angle T=180^{\circ}\), so \(m\angle S + m\angle T=180^{\circ}-m\angle R\).
Given \(m\angle R>m\angle S + m\angle T\), substitute \(m\angle S + m\angle T\) with \(180 - m\angle R\). Then \(m\angle R>180 - m\angle R\).
Add \(m\angle R\) to both sides: \(m\angle R+m\angle R>180^{\circ}\), \(2m\angle R>180^{\circ}\), \(m\angle R > 90^{\circ}\).

Step2: Analyze the relationship between angles

Since \(m\angle R>m\angle S + m\angle T\), and \(m\angle S\geq0\), \(m\angle T\geq0\).
If \(m\angle R>m\angle S + m\angle T\), then \(m\angle R>m\angle T\) (because \(m\angle S\geq0\)) and \(m\angle R>m\angle S\) (because \(m\angle T\geq0\))

Answer:

\(m\angle R>m\angle T\), \(m\angle R>m\angle S\), \(m\angle R > 90^{\circ}\)