Question

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

Explanation:

To prove \(\triangle ABC \sim \triangle TRS\) using the AA (Angle - Angle) similarity theorem, we need to show that two pairs of corresponding angles are equal.

1. First, we know that in \(\triangle ABC\), we are given \(m\angle B = 46^{\circ}\) and \(m\angle C=45^{\circ}\). To find \(m\angle A\), we use the fact that the sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle A=180^{\circ}-m\angle B - m\angle C=180^{\circ}-46^{\circ}-45^{\circ} = 89^{\circ}\).
2. In \(\triangle TRS\), we are given \(m\angle R = 46^{\circ}\) and \(m\angle T = 89^{\circ}\). If we know that the sum of the interior angles of a triangle is \(180^{\circ}\), we can find \(m\angle S=180^{\circ}-m\angle R - m\angle T=180^{\circ}-46^{\circ}-89^{\circ}=45^{\circ}\).
3. Now we can see that \(m\angle B=m\angle R = 46^{\circ}\) and \(m\angle A=m\angle T = 89^{\circ}\) (or \(m\angle C=m\angle S = 45^{\circ}\)), so by AA similarity, the triangles are similar.

The other options are not relevant:

• Saying \(\triangle ABC\) is an acute triangle does not help in proving similarity.
• The size of the triangle (whether \(\triangle TRS\) is larger than \(\triangle ABC\)) has no bearing on similarity (similarity is about shape, not size).
• The triangle inequality (sum of two sides greater than the third side) is related to the existence of a triangle, not similarity.

Answer:

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