To solve the problem, we need to analyze the geometric relationships between the points and angles given in the question. Let's break it down step-by-step. ### Step 1: Understanding the Geometry 1. **Identify the Points and Lines**: - We have a straight line UV. - Point P is on one side of line UV and moves in the same direction as UV. - There is another line ST that represents the locus of point P. - The angle VUS is given as 50°. 2. **Visualizing the Setup**: - Imagine line UV as a horizontal line. - Point U is on the left, and point V is on the right. - The angle VUS is formed at point U, where line UV meets line ST. ### Step 2: Analyzing the Angles 1. **Understanding Angle VUS**: - Angle VUS = 50° means that if you were to draw a line from U to V and then from U to S, the angle formed between these two lines at point U is 50°. 2. **Finding Angle UST**: - We need to find angle UST, which is the angle formed between line US and line ST. ### Step 3: Using the Properties of Angles 1. **Exterior Angle Theorem**: - In any triangle, the exterior angle is equal to the sum of the two opposite interior angles. However, in this case, we can use the fact that angles around a point sum up to 360°. 2. **Calculating Angle UST**: - Since angle VUS is 50°, we can find angle UST by considering the angles around point U. - The angles around point U can be expressed as: \[ \text{Angle VUS} + \text{Angle UST} + \text{Angle STU} = 360° \] - However, we need to recognize that angle STU is actually the same as angle VUS because both lines (UV and ST) are extending in the same direction. Therefore, angle STU is also 50°. 3. **Setting Up the Equation**: - We can now set up the equation: \[ 50° + \text{Angle UST} + 50° = 360° \] - Simplifying this gives: \[ 100° + \text{Angle UST} = 360° \] - Therefore: \[ \text{Angle UST} = 360° - 100° = 260° \] ### Step 4: Correcting the Calculation 1. **Revisiting the Angles**: - We realize that angle UST is actually the angle on the opposite side of angle VUS. Thus, we should consider the supplementary angle: \[ \text{Angle UST} = 180° - 50° = 130° \] ### Step 5: Evaluating the Options - **Correct Option**: B. 130° - **Why Other Options Are Incorrect**: - **A. 310°**: This angle is not relevant in this context as it exceeds 360° and does not fit the geometric configuration. - **C. 80°**: This angle does not correspond to any of the angles derived from the given information. - **D. 50°**: This is the angle VUS, not UST. ### Summary - The angle UST is calculated using the relationship between angles around point U. - The correct answer is B. 130°. - The other options do not fit the geometric relationships established in the problem. ### Revision Summary - Understand the relationships between angles in geometric configurations. - Use the properties of angles around a point to find unknown angles. - Remember that supplementary angles can help in finding angles that are not directly given. - Always visualize the problem to better understand the relationships between points and lines.