Given that $m\angle 1=m\angle 4$, $m\angle 2=m\angle 3$, $m\angle 1+m\angle 4=180°$, and $m\angle 2+m\angle 3=180°$, which of the following best describes the relationship between the angles in two triangles that allows the Law of Sines to be applied?

• A. The triangles are right triangles because each has a $90°$ angle.
• B. The triangles are similar because their corresponding angles are equal.
• C. The triangles are congruent because all their sides are equal.
• D. The triangles are isosceles because two angles in each triangle are equal.

Answer: (B)

Verified step by step guidance

1. Identify the given information: \( m\angle 1 = m\angle 4 \), \( m\angle 2 = m\angle 3 \), \( m\angle 1 + m\angle 4 = 180^\circ \), and \( m\angle 2 + m\angle 3 = 180^\circ \). This tells us that the pairs of angles are equal and supplementary in their respective triangles.
2. Recall that the Law of Sines applies to any triangle, but it is especially useful when the triangles involved are similar, meaning their corresponding angles are equal and their sides are proportional.
3. Since \( m\angle 1 = m\angle 4 \) and \( m\angle 2 = m\angle 3 \), the two triangles have two pairs of equal corresponding angles. By the Angle-Angle (AA) similarity criterion, this means the triangles are similar.
4. Understand that similarity of triangles implies that their corresponding sides are proportional, which is the key condition that allows the Law of Sines to be applied between the two triangles.
5. Therefore, the best description of the relationship between the angles in the two triangles that allows the Law of Sines to be applied is that the triangles are similar because their corresponding angles are equal.