find the measure of the indicated angle to the nearest degree.

1. For problem 7:
   • Recall the tangent - function in a right - triangle. The tangent of an acute angle in a right - triangle is defined as the ratio of the opposite side to the adjacent side. Let the indicated angle be (\theta). Here, the opposite side (a = 51) and the adjacent side (b = 55).
   • Step 1: Write the tangent formula
     • (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{51}{55}).
   • Step 2: Find the angle using the inverse - tangent function
     • (\theta=\tan^{- 1}(\frac{51}{55})). Using a calculator, (\theta=\tan^{-1}(0.9273)\approx43^{\circ}).
2. For problem 8:
   • The opposite side (a = 19) and the adjacent side (b = 27).
   • Step 1: Write the tangent formula
     • (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{19}{27}).
   • Step 2: Find the angle using the inverse - tangent function
     • (\theta=\tan^{-1}(\frac{19}{27})). Using a calculator, (\theta=\tan^{-1}(0.7037)\approx35^{\circ}).
3. For problem 9:
   • The opposite side (a = 34) and the adjacent side (b = 55).
   • Step 1: Write the tangent formula
     • (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{34}{55}).
   • Step 2: Find the angle using the inverse - tangent function
     • (\theta=\tan^{-1}(\frac{34}{55})). Using a calculator, (\theta=\tan^{-1}(0.6182)\approx32^{\circ}).
4. For problem 10:
   • The opposite side (a = 14) and the adjacent side (b = 24).
   • Step 1: Write the tangent formula
     • (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{14}{24}=\frac{7}{12}).
   • Step 2: Find the angle using the inverse - tangent function
     • (\theta=\tan^{-1}(\frac{7}{12})). Using a calculator, (\theta=\tan^{-1}(0.5833)\approx30^{\circ}).
5. For problem 11:
   • The opposite side (a = 29) and the adjacent side (b = 38).
   • Step 1: Write the tangent formula
     • (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{29}{38}).
   • Step 2: Find the angle using the inverse - tangent function
     • (\theta=\tan^{-1}(\frac{29}{38})). Using a calculator, (\theta=\tan^{-1}(0.7632)\approx37^{\circ}).
6. For problem 12:
   • The opposite side (a = 8) and the adjacent side (b = 20).
   • Step 1: Write the tangent formula
     • (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{8}{20}=0.4).
   • Step 2: Find the angle using the inverse - tangent function
     • (\theta=\tan^{-1}(0.4)). Using a calculator, (\theta=\tan^{-1}(0.4)\approx22^{\circ}).
7. For problem 13:
   • The opposite side (a = 34) and the adjacent side (b = 42).
   • Step 1: Write the tangent formula
     • (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{34}{42}=\frac{17}{21}).
   • Step 2: Find the angle using the inverse - tangent function
     • (\theta=\tan^{-1}(\frac{17}{21})). Using a calculator, (\theta=\tan^{-1}(0.8095)\approx39^{\circ}).
8. For problem 14:
   • The opposite side (a = 52) and the adjacent side (b = 56).
   • Step 1: Write the tangent formula
     • (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{52}{56}=\frac{13}{14}).
   • Step 2: Find the angle using the inverse - tangent function
     • (\theta=\tan^{-1}(\frac{13}{14})). Using a calculator, (\theta=\tan^{-1}(0.9286)\approx43^{\circ}).

Answer:

7. (43^{\circ})
8. (35^{\circ})
9. (32^{\circ})
10. (30^{\circ})
11. (37^{\circ})
12. (22^{\circ})
13. (39^{\circ})
14. (43^{\circ})