Final answer:
The number of triangles that can be made with two side lengths and a non-included angle depends on the relationship between the sides and the given angle, based on the Law of Sines. One or two triangles may be possible, or none, if no triangle can satisfy the given conditions.
Step-by-step explanation:
The question pertains to the number of different triangles that can be made with two given side lengths and an angle that is not between those sides, known as the Side-Side-Angle (SSA) condition. To determine the number of possible triangles, one must consider the Law of Sines. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles in a triangle.
Given side lengths of 6 cm and 7 cm, and an angle of 35°, we need to consider the following:
- If the given angle is opposite the longer side, there is exactly one possible triangle.
- If the given angle is opposite the shorter side, there could be two cases:
- There is one possible triangle if the height corresponding to the longer side is less than the length of the shorter side for the constructed right triangle.
- There are two possible triangles if the height is greater than the shorter side but less than the longer side.
- There are no possible triangles if the given angle is acute and opposite the shorter side, and the longer side is shorter than the shorter side divided by the sine of the given angle.
However, without additional information, the exact number of triangles cannot be determined, and we need to use additional geometric or trigonometric properties to solve for specific cases.