The order of the side lengths from smallest to largest is UV, U, and VU, and the order of the angle measures from smallest to largest is angle UV, angle U, and angle VU. Also the value of angle T is 60°.
Based on the triangle inequality, we know that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have:
UV + VU > 58
UV + 58 > 62
VU + 58 > 62
From these inequalities, we can see that UV is the shortest side of the triangle.
The angle opposite the shortest side of a triangle is also the smallest angle. Therefore, angle UV is the smallest angle in the triangle.
Similarly, we can use the triangle inequality to determine that VU is the longest side of the triangle. Therefore, angle VU is the largest angle in the triangle.
Finally, the remaining angle, angle U, must be between angle UV and angle VU.
Therefore, the order of the side lengths from smallest to largest is UV, U, and VU, and the order of the angle measures from smallest to largest is angle UV, angle U, and angle VU.
The value of angle T is 60 degrees.
Identify the relevant angles: We are given that angle U measures 58 degrees and angle V measures 62 degrees. We need to find the measure of angle T.
Recognize the triangle: The problem doesn't explicitly mention a triangle, but the three angles U, V, and T together form a triangle UVT.
Use the angle sum property of triangles: The sum of the angles in a triangle is always 180 degrees. Therefore, in triangle UVT, we have:
angle U + angle V + angle T = 180 degrees
Substitute the known values and solve for angle T: We know angle U = 58 degrees and angle V = 62 degrees. Plugging these values into the equation, we get:
58 degrees + 62 degrees + angle T = 180 degrees
Combining like terms, we get:
120 degrees + angle T = 180 degrees
Subtracting 120 degrees from both sides, we get:
angle T = 180 degrees - 120 degrees
Therefore, angle T = 60 degrees.