Question

given ( mangle b = 46^{circ}), ( mangle c = 45^{circ}), ( mangle r = 46^{circ}), ( mangle t = 89^{circ}). prove (\triangle abcsim\triangle trs). melissa believes that the aa similarity theorem can prove that the triangles are similar. which fact would be necessary in the proof? (\triangle abc) is an acute triangle. (\triangle trs) is larger than (\triangle abc). the sum of the measures of the interior angles of a triangle is ( 180^{circ}). the sum of the side lengths of two sides of a triangle is greater than the third side length.

Explanation:

To use the AA (Angle - Angle) similarity theorem, we need to show that two angles of one triangle are equal to two angles of another triangle. For \(\triangle ABC\), we know \(m\angle B = 46^{\circ}\) and \(m\angle C=45^{\circ}\). Using the fact that the sum of interior angles of a triangle is \(180^{\circ}\), we can find \(m\angle A=180^{\circ}-(46^{\circ} + 45^{\circ})=89^{\circ}\). For \(\triangle TRS\), \(m\angle R = 46^{\circ}\) and \(m\angle T = 89^{\circ}\). Then, using the angle - sum property again, \(m\angle S=180^{\circ}-(46^{\circ}+89^{\circ}) = 45^{\circ}\). So, \(\angle B=\angle R = 46^{\circ}\) and \(\angle A=\angle T = 89^{\circ}\), which allows us to apply the AA similarity theorem. The other options (A: triangle type, B: size comparison, D: triangle side - length property) are not relevant for proving similarity via the AA theorem.

Answer:

C. The sum of the measures of the interior angles of a triangle is \(180^{\circ}\)