Question

Lakie is given two side lengths and the measure of the angle between them. How many unique triangles can she form given this information?

Answer 1

Given two side lengths and the measure of the angle between them, Lakie can typically form one unique triangle, known as SAS configuration. However, in some cases with an obtuse or right angle and particular side lengths, multiple triangles or no triangle at all could be possible.

Step-by-step explanation:

When Lakie is given two side lengths and the measure of the angle between them, the triangle that can be formed from this information is typically unique, resulting in one specific triangle. This configuration is known in geometry as SAS (Side-Angle-Side), which is one of the ways to prove that two triangles are congruent if all three properties match. Since the side lengths and the angle fix the size and shape of the triangle, the proportions and angles of the resulting triangle will always be the same, making it unique. The only exception is in a specific case when the known angle is obtuse or right and the opposite side is longer than the other given side, which could lead to different triangles or no triangle at all due to the Law of Sines.

Answer 2

She can form only one such triangle. The given angle 'fixes' the shape of the triangle. Once the angle is given, the third side is also fixed.

So we have the first 2 sides, say a and b, and the third side c, fixed by fixing the angle between a and b. By Side Side Side congruence postulate, any triangle with sides a, b, c will be the exact same triangle as the one we constructed.