Answer:To find TU and mZOSR, let's start by drawing a diagram.
The diagram would look like this:
```
O__________R
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
T____________________________________U
\ /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
S
|
|
Q
```
Since T is the midpoint of OR, we can say that RT = TO. Similarly, since U is the midpoint of QS, we can say that SQ = QU.
Since R$ = 38, we can say that RT = TO = 38/2 = 19.
Since m LQUT = 75°, we know that angles LQU and UTQ are equal and both measure 75°.
Now, let's find TU. We know that TU = QT - QU.
Since QT is the sum of angles UTQ and UTS, and we know that angle UTQ = 75°, we need to find angle UTS.
Since angle LQU = angle UTQ, and we know that angle LQU = 75°, we can subtract 75° from 180° to find angle UTS.
Angle UTS = 180° - 75° = 105°.
Now, we can find QT by subtracting angle UTS from 180°.
QT = 180° - 105° = 75°.
Finally, we can find TU by subtracting QU from QT.
TU = QT - QU = 75° - SQ = 75° - 19 = 56°.
So, TU measures 56°.
Now, let's find mZOSR.
Since T is the midpoint of OR, we know that angle ZOT is a right angle (90°). Similarly, since U is the midpoint of QS, we know that angle ZUS is also a right angle (90°).
Since angles ZOT and ZUS are vertical angles, they are equal. Therefore, angle ZOT = angle ZUS = 90°.
Now, let's find angle ZOS.
Angle ZOS is the sum of angles ZOT and ZUS.
Angle ZOS = angle ZOT + angle ZUS = 90° + 90° = 180°.
Therefore, mZOSR = 180°.
To summarize:
- TU measures 56°.
- mZOSR measures 180°.
Step-by-step explanation: Can you give me 100 points