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We are given a triangle \( \triangle ABC \) with the following information:

- \( A = 67^\circ \)
- \( C = 56^\circ \)
- Side \( a = 70 \)

We need to solve the triangle, meaning we have to find the remaining side \( b \), side \( c \), and angle \( B \).

### Step 1: Find angle \( B \)
Since the sum of the angles in any triangle is \( 180^\circ \), we can find angle \( B \) using:

\[
B = 180^\circ - A - C = 180^\circ - 67^\circ - 56^\circ = 57^\circ
\]

### Step 2: Apply the Law of Sines
The Law of Sines states that:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

We can use this to find sides \( b \) and \( c \).

### Step 3: Solve for side \( b \)
Using the Law of Sines:

\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]

Substituting known values:

\[
\frac{70}{\sin 67^\circ} = \frac{b}{\sin 57^\circ}
\]

First, calculate the sine values:

- \( \sin 67^\circ \approx 0.9205 \)
- \( \sin 57^\circ \approx 0.8387 \)

Now, solve for \( b \):

\[
\frac{70}{0.9205} = \frac{b}{0.8387}
\]

\[
b = \frac{70 \times 0.8387}{0.9205} \approx 63.7
\]

### Step 4: Solve for side \( c \)
Similarly, we use the Law of Sines to solve for \( c \):

\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]

Substitute the known values:

\[
\frac{70}{\sin 67^\circ} = \frac{c}{\sin 56^\circ}
\]

Using \( \sin 56^\circ \approx 0.8290 \), we solve for \( c \):

\[
\frac{70}{0.9205} = \frac{c}{0.8290}
\]

\[
c = \frac{70 \times 0.8290}{0.9205} \approx 63.0
\]

### Final Solution:
- \( B = 57^\circ \)
- \( b \approx 63.7 \)
- \( c \approx 63.0 \)

Would you like further details or have any questions?

Here are some related questions you might explore:
1. How does the Law of Sines apply to obtuse triangles?
2. What are the key differences between the Law of Sines and the Law of Cosines?
3. Can a triangle with the given angles and sides have a different configuration?
4. How would the solution change if one of the angles was obtuse?
5. How do you solve a triangle when given two sides and a non-included angle?

**Tip:** When solving triangles, always ensure the angle measures add up to \(180^\circ\) to avoid calculation errors.

Consider a triangle ABC where A = 67°, C = 56°, and a = 70. Solve the triangle.

Learn how to solve triangle ABC, where A = 67°, C = 56°, and side a = 70, using the Law of Sines. This step-by-step solution calculates the remaining angles and sides, suitable for high school trigonometry students.

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