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: Recall triangle angle sum

In any triangle, the sum of interior angles is \(180^\circ\). So for \(\triangle RST\), \(m\angle R + m\angle S + m\angle T = 180^\circ\), which implies \(m\angle S + m\angle T = 180^\circ - m\angle R\).

Step2: Analyze \(m\angle R > m\angle S + m\angle T\)

Substitute \(m\angle S + m\angle T = 180^\circ - m\angle R\) into \(m\angle R > m\angle S + m\angle T\). We get \(m\angle R > 180^\circ - m\angle R\).

Step3: Solve for \(m\angle R\)

Add \(m\angle R\) to both sides: \(2m\angle R > 180^\circ\), then divide by 2: \(m\angle R > 90^\circ\). So the first option is true.

Step4: Analyze \(m\angle S + m\angle T\)

Since \(m\angle R > 90^\circ\) and \(m\angle R + m\angle S + m\angle T = 180^\circ\), then \(m\angle S + m\angle T = 180^\circ - m\angle R < 180^\circ - 90^\circ = 90^\circ\). So the second option is true.

Step5: Analyze \(m\angle R > m\angle T\) and \(m\angle R > m\angle S\)

We know \(m\angle R > m\angle S + m\angle T\). Since \(m\angle S\) and \(m\angle T\) are positive angles, \(m\angle R > m\angle S\) (because \(m\angle S + m\angle T > m\angle S\)) and \(m\angle R > m\angle T\) (because \(m\angle S + m\angle T > m\angle T\)). So the fourth and fifth options are true.

Step6: Analyze \(m\angle S = m\angle T\) and \(m\angle S > m\angle T\)

There's no information given to suggest \(m\angle S = m\angle T\) or \(m\angle S > m\angle T\). These depend on the specific triangle, not the given inequality \(m\angle R > m\angle S + m\angle T\). So these options are not necessarily true.

Answer:

• \(m\angle R > 90^\circ\)
• \(m\angle S + m\angle T < 90^\circ\)
• \(m\angle R > m\angle T\)
• \(m\angle R > m\angle S\)