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To determine how many solutions this triangle has, we need to apply the Law of Sines and check for any ambiguities. Specifically, we need to see if the triangle could have 0, 1, or 2 solutions.

Given:
- \( m \angle A = 70^\circ \)
- \( b = 26 \)
- \( a = 25 \)

Since we know one angle and two sides, we can use the **Ambiguous Case** for the SSA (Side-Side-Angle) configuration, which can have:
1. No solution.
2. One solution.
3. Two solutions.

### Step 1: Calculate \( \sin B \)
Using the Law of Sines:
\[
\frac{\sin A}{a} = \frac{\sin B}{b}
\]
Plugging in the values we know:
\[
\frac{\sin 70^\circ}{25} = \frac{\sin B}{26}
\]
Solving for \( \sin B \):
\[
\sin B = 26 \times \frac{\sin 70^\circ}{25}
\]

Calculating this:

1. \( \sin 70^\circ \approx 0.9397 \)
2. \( \sin B \approx 26 \times \frac{0.9397}{25} \approx 0.976 \)

Since \( \sin B \approx 0.976 \), and this value is less than 1, \( B \) exists. We now check if there can be one or two solutions.

### Step 2: Determine Possible Angles for \( B \)
1. Calculate the first possible \( B \):
   \[
   B = \arcsin(0.976) \approx 78.95^\circ
   \]

2. The second possible angle for \( B \) would be \( B' = 180^\circ - 78.95^\circ = 101.05^\circ \).

### Step 3: Check for Validity of Each Solution
1. **For \( B \approx 78.95^\circ \):**
   - The third angle \( C = 180^\circ - 70^\circ - 78.95^\circ \approx 31.05^\circ \).
   - This is a valid solution since all angles add up to \( 180^\circ \).

2. **For \( B' \approx 101.05^\circ \):**
   - The third angle \( C = 180^\circ - 70^\circ - 101.05^\circ \approx 8.95^\circ \).
   - This is also a valid solution since all angles add up to \( 180^\circ \).

### Conclusion
Since both configurations are valid, this triangle has **two solutions**.

Would you like further details on any of these steps?

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### Related Questions
1. How does the Law of Sines help solve triangles?
2. What is the Ambiguous Case in trigonometry, and when does it apply?
3. How do you determine the number of solutions in an SSA triangle?
4. What happens if \( \sin B > 1 \) when applying the Law of Sines?
5. Can the Law of Sines be used to find angles in a right triangle?

### Tip
For SSA triangles, remember to check if the opposite side is shorter, longer, or equal to the height from the angle to determine possible solutions.

Given m∠A = 70°, b = 26, and a = 25, determine how many solutions this triangle has.

Explore how to determine the number of possible solutions for a triangle with angle m∠A = 70° and sides b = 26 and a = 25. Learn about the Law of Sines and the Ambiguous Case (SSA) with a step-by-step analysis suitable for high school students.

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