ABC is a triangle in which 35 times the smallest angle is equal to the 26 times largest angle. What is the measure of the second largest angle?

Options

A. `63^(@)`
B. `58^(@)`
C. `70^(@)`
D. `42^(@)`

Answer

To solve the problem step by step, we will first define the angles of the triangle and then use the given information to find the measure of the second largest angle.

### Step 1: Define the angles
Let:
- Angle A = smallest angle
- Angle B = second largest angle
- Angle C = largest angle

### Step 2: Set up the equation based on the given condition
According to the problem, we have:
\[ 35 \times \text{(smallest angle)} = 26 \times \text{(largest angle)} \]
This translates to:
\[ 35A = 26C \]
We can rearrange this to:
\[ 35A - 26C = 0 \]
This is our first equation.

### Step 3: Express angles in terms of a variable
Let’s express the angles in terms of a variable \( K \):
- Assume \( A = 26K \) (smallest angle)
- Assume \( C = 35K \) (largest angle)

### Step 4: Use the triangle angle sum property
The sum of the angles in a triangle is always 180 degrees:
\[ A + B + C = 180^\circ \]
Substituting the expressions for A and C:
\[ 26K + B + 35K = 180^\circ \]
This simplifies to:
\[ B + 61K = 180^\circ \]
Thus, we can express B as:
\[ B = 180^\circ - 61K \]

### Step 5: Determine the order of angles
Since \( A \) is the smallest angle and \( C \) is the largest angle, we need to ensure:
1. \( A < B < C \)
2. \( 26K < 180^\circ - 61K < 35K \)

### Step 6: Solve the inequalities
1. From \( 26K < 180^\circ - 61K \):
\[ 26K + 61K < 180^\circ \]
\[ 87K < 180^\circ \]
\[ K < \frac{180^\circ}{87} \approx 2.07 \]

2. From \( 180^\circ - 61K < 35K \):
\[ 180^\circ < 35K + 61K \]
\[ 180^\circ < 96K \]
\[ K > \frac{180^\circ}{96} \approx 1.875 \]

### Step 7: Determine the integer value of K
The integer value of \( K \) must satisfy:
\[ 1.875 < K < 2.07 \]
Thus, the only integer value for \( K \) is \( K = 2 \).

### Step 8: Calculate the angles
Now substituting \( K = 2 \) back into the expressions for the angles:
- \( A = 26K = 26 \times 2 = 52^\circ \)
- \( C = 35K = 35 \times 2 = 70^\circ \)
- Now, substituting \( K \) into the equation for \( B \):
\[ B = 180^\circ - 61 \times 2 = 180^\circ - 122^\circ = 58^\circ \]

### Conclusion
The measure of the second largest angle \( B \) is:
\[ \boxed{58^\circ} \]