Question

given: m∠b = 46°; m∠c = 45°; m∠r = 46°; m∠t = 89° prove: △abc ~ △trs melissa believes that the aa similarity theorem can prove that the triangles are similar. which fact would be necessary in the proof? ○ △abc is an acute triangle. ○ △trs is larger than △abc. ○ the sum of the measures of the interior angles of a triangle is 180°. ○ 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 congruent to two angles of another triangle. For $\triangle ABC$, we know $m\angle B = 46^{\circ}$ and $m\angle C=45^{\circ}$. We can find $m\angle A$ using the fact that the sum of the interior angles of a triangle is $180^{\circ}$. So, $m\angle A=180^{\circ}-m\angle B - m\angle C=180 - 46-45 = 89^{\circ}$. In $\triangle TRS$, $m\angle R = 46^{\circ}$ and $m\angle T = 89^{\circ}$. Now we can see that $\angle B\cong\angle R$ (both $46^{\circ}$) and $\angle A\cong\angle T$ (both $89^{\circ}$), which allows us to apply the AA similarity theorem.

• The option " $\triangle ABC$ is an acute triangle" is not necessary for proving similarity.
• The option " $\triangle TRS$ is larger than $\triangle ABC$" is irrelevant to the AA similarity theorem.
• The option "The sum of the side lengths of two sides of a triangle is greater than the third side length" is the triangle inequality theorem, which is used for triangle construction, not similarity.

Answer:

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