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Solve ΔABCΔABC . Give the measures to the nearest tenth where necessary.

 ∠A=68∠A=68 degrees

 ∠B=∠B=  degrees

 ∠C=90∠C=90 degrees

  a=10a=10

  b=b= 

  c=c= 

To solve triangle \( \Delta ABC \) with the given information:

1. **Angles:**
   - \( \angle A = 68^\circ \)
   - \( \angle C = 90^\circ \) (since it's a right triangle)
   - Using the fact that the sum of angles in a triangle is \( 180^\circ \), we find \( \angle B \):
     \[
     \angle B = 180^\circ - \angle A - \angle C = 180^\circ - 68^\circ - 90^\circ = 22^\circ
     \]

2. **Sides:**
   - Given \( a = 10 \) (side opposite \( \angle A \)), we need to find sides \( b \) (opposite \( \angle B \)) and \( c \) (the hypotenuse opposite \( \angle C \)).

3. **Finding \( b \) (using the sine function):**
   \[
   \frac{a}{\sin A} = \frac{b}{\sin B} \implies b = a \cdot \frac{\sin B}{\sin A}
   \]
   \[
   b = 10 \cdot \frac{\sin(22^\circ)}{\sin(68^\circ)}
   \]
   \[
   \sin(22^\circ) \approx 0.3746, \quad \sin(68^\circ) \approx 0.9272
   \]
   \[
   b \approx 10 \cdot \frac{0.3746}{0.9272} \approx 4.0
   \]

4. **Finding \( c \) (using the cosine function):**
   \[
   c = \frac{a}{\sin A} = \frac{10}{\sin(68^\circ)} \approx \frac{10}{0.9272} \approx 10.8
   \]

### Summary of the Results:
- \( \angle A = 68^\circ \)
- \( \angle B = 22^\circ \)
- \( \angle C = 90^\circ \)
- \( a = 10 \)
- \( b \approx 4.0 \)
- \( c \approx 10.8 \)

Would you like to know more details or have any questions about these calculations? Here are some related questions for further exploration:

1. How do the sine and cosine laws apply in different types of triangles?
2. What is the significance of the angles in triangle properties?
3. How can you calculate the area of triangle \( \Delta ABC \) using these sides?
4. What would happen to the side lengths if \( a \) were increased?
5. Can you explain the relationship between the angles and the sides in a right triangle? 

**Tip:** Always use a scientific calculator or a trigonometric table for precise values of sine and cosine when solving triangle problems.

Solve ΔABC given ∠A=68°, ∠C=90°, a=10, find ∠B, b, and c.

Explore the solution for triangle ABC where ∠A=68°, ∠C=90°, and side a=10. Discover how to find the missing angle B and sides b and c using trigonometric principles. This guide is perfect for high school students studying trigonometry.

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