The measure of angle TSU is 80 degrees. In the most common version of this problem, angle TSU forms a linear pair with angle VSU, which measures 100 degrees. Since linear pairs are supplementary and add up to 180 degrees, angle TSU equals 180 minus 100, giving you 80 degrees.
Why Angles TSU and VSU Add to 180
A linear pair is created when two angles sit next to each other and their non-shared sides form a straight line. A straight line measures exactly 180 degrees, so the two angles must combine to reach that total. In this problem, rays ST and SV extend in opposite directions from point S, with ray SU between them. That makes angles TSU and VSU a linear pair.
The calculation is straightforward:
- Angle VSU = 100°
- Angle TSU = 180° − 100° = 80°
How This Works With Different Values
Some versions of this problem use different numbers for angle VSU. The method stays the same: subtract the given angle from 180. If angle VSU were 120 degrees, angle TSU would be 60 degrees. If angle VSU were 70 degrees, angle TSU would be 110 degrees. The relationship between the two angles never changes as long as they form a linear pair.
Recognizing Supplementary Angles
Supplementary angles are any two angles that add up to 180 degrees. A linear pair is one specific way supplementary angles show up in geometry, but two angles don’t need to be adjacent to be supplementary. The key distinction is that a linear pair always involves two angles sharing a vertex and a common side, with their outer sides forming a straight line. When you spot that arrangement in a diagram, you immediately know the two angles sum to 180.
This concept appears frequently in problems involving intersecting lines, triangles, and polygons. For triangles, the interior angles add up to 180 degrees on their own. When a triangle’s side is extended beyond a vertex, the angle formed outside the triangle creates a linear pair with the interior angle at that vertex. So if you know the exterior angle, you can find the interior angle the same way: subtract from 180.
Solving Without a Diagram
If your version of the problem includes a diagram you’re trying to interpret, look for these clues. First, check whether points T, S, and V appear to lie along a straight line with U off to one side. That confirms the linear pair. Second, find the angle measurement given in the problem, which is typically attached to angle VSU. Then subtract that value from 180 to get angle TSU.
If the diagram instead shows a triangle with vertices T, S, and U, you would use the triangle angle sum property. All three interior angles of any triangle add to 180 degrees, so knowing two of the three angles lets you solve for the third. For the standard version of this problem, though, the linear pair approach is what gets you to the answer.