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 two pairs of congruent angles. We know \(m\angle B = 46^{\circ}\) and \(m\angle R=46^{\circ}\). For the second pair of angles, we need to find the measure of the third angle in \(\triangle ABC\) and \(\triangle TRS\). The sum of the interior angles of a triangle is \(180^{\circ}\). In \(\triangle ABC\), \(m\angle A=180^{\circ}-(46^{\circ} + 45^{\circ})=89^{\circ}\). In \(\triangle TRS\), if we use the angle - sum property (\(m\angle S=180^{\circ}-(46^{\circ}+89^{\circ}) = 45^{\circ}\)), we can get the second pair of congruent angles (\(\angle C\) and \(\angle S\) or \(\angle A\) and \(\angle T\)).

Answer:

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