Answer:
a 63° angle between two sides of length 4 cm and 7 cm.
Explanation:
A triangle can only be formed if the sum of any two sides is greater than the third side, and in the case of angles, the sum of the angles must be equal to 180 degrees. Using these principles, we can determine which measure would create only one possible triangle:
- three angles of 39°, 68°, and 69°: This set of angles does not form a triangle because the sum of the angles is 176°, which is less than the required 180°. Therefore, no triangle can be formed with these angles.
- two sides of 4 cm and 7 cm with an angle of 54° not between those two sides: This set of measures can form two possible triangles. To see why, imagine drawing a line segment of length 7 cm, then drawing two circles centered at the endpoints with radii of 4 cm. The two circles will intersect at two points, and these points represent the two possible locations for the third vertex of the triangle. Therefore, there are two possible triangles that can be formed with these measures.
- a 63° angle between two sides of length 4 cm and 7 cm: This set of measures can only form one possible triangle. To see why, note that the angle opposite the side of length 4 cm must be less than 63° (since the sum of angles in a triangle is 180°). Likewise, the angle opposite the side of length 7 cm must be greater than 63°. Therefore, there is only one possible location for the third vertex of the triangle, and hence only one possible triangle can be formed with these measures.