Question

in triangle rst, ( mangle r>mangle s + mangle t ). which must be true of triangle rst? check all that apply.
( square mangle r>90^{circ} )
( square mangle s + mangle t<90^{circ} )
( square mangle s=mangle t )
( square mangle r>mangle t )
( square mangle r>mangle s )
( square mangle s>mangle t )

Explanation:

Step1: Use the triangle - angle sum property

In a triangle \(RST\), \(m\angle R+m\angle S + m\angle T=180^{\circ}\). Given \(m\angle R>m\angle S + m\angle T\). Let \(x = m\angle S + m\angle T\), then \(m\angle R>x\) and \(m\angle R+x = 180^{\circ}\). Substituting \(x\) with \(m\angle R\) (since \(m\angle R>x\)), we get \(m\angle R+m\angle R>180^{\circ}\) (because if \(a > b\) and \(a + b=180^{\circ}\), then \(2a>180^{\circ}\)). So \(2m\angle R>180^{\circ}\), which simplifies to \(m\angle R > 90^{\circ}\).

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 \(m\angle S\geq0\), so \(m\angle S + m\angle T\geq m\angle T\)) and \(m\angle R>m\angle S\) (because \(m\angle S + m\angle T\geq m\angle S\)).

Answer:

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