To solve this problem using the Law of Sines, let's break down the information given in the triangle $\triangle ABC$:

1. Given Information:

   • Angle $A = 63^\circ$
   • Side $a = 23$
   • Side $c = 11$

2. Objective:

   • Solve for side $b$, angle $B$, and angle $C$.
   • Round answers to the nearest tenth.

Step 1: Using the Law of Sines

The Law of Sines states that: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Step 2: Find Angle $C$

Since $A$ and $C$ are opposite sides $a$ and $c$, we can write: $\frac{23}{\sin 63^\circ} = \frac{11}{\sin C}$ Solving for $\sin C$: $\sin C = \frac{11 \cdot \sin 63^\circ}{23}$ Calculate $\sin C$ to find the value of $C$.

Step 3: Find Angle $B$

Once we have $C$, use the fact that the sum of angles in a triangle is $180^\circ$: $B = 180^\circ - A - C$

Step 4: Find Side $b$

Finally, use the Law of Sines to find $b$: $\frac{b}{\sin B} = \frac{23}{\sin 63^\circ}$ Rearrange to solve for $b$: $b = \frac{23 \cdot \sin B}{\sin 63^\circ}$

With these steps, you can solve the triangle and check against the answer choices provided.

Would you like the detailed calculations for each step, or do you have questions about a specific part?

Here are five related questions:

1. What is the Law of Cosines, and how is it used in solving triangles?
2. How can we confirm whether a triangle is solvable with given side and angle measures?
3. How does the sum of angles in a triangle affect calculations in non-right triangles?
4. Why is it important to use radians or degrees consistently in trigonometric calculations?
5. How can ambiguous cases occur in Law of Sines calculations?

Tip: Always verify if your triangle’s dimensions satisfy the given conditions, especially in cases with obtuse angles.