To find the measures of angles $\angle QRT$ and $\angle QTR$, let's analyze the given data and assume it relates to a straight line or polygon structure.

Step 1: Assign the Line Segments

We are given the measures of segments:

• $m_{QU} = 78$
• $m_{UT} = 38$
• $m_{TS} = 56$
• $m_{RS} = 76$

Let’s assume these measurements define points $Q$, $U$, $T$, $S$, and $R$ placed sequentially on a line or a polygon.

Step 2: Evaluate Geometry and Angles

The specific structure (line or polygon) affects the interpretation:

1. If the points form a straight line:
   The total length of the line is the sum of these segments. For angle measures:

   • Angles $\angle QRT$ and $\angle QTR$ would depend on the geometry created by intersecting lines or external conditions.

2. If the points form a polygon:
   Typically, you would calculate interior angles using the segments as side lengths or through trigonometric relationships.

Step 3: Missing Information

To compute $\angle QRT$ and $\angle QTR$ precisely, I need additional details such as:

• Are the points forming a triangle or some other polygon?
• Are there any right angles, parallel lines, or other constraints?

Please clarify or provide a diagram for further calculations! Meanwhile, here’s a tip:

Tip:

Visualizing the problem or providing a rough sketch can help identify key geometric relationships and assumptions.

Let me know if you need further help or have a clarification!