The side BC of a triangle ABC is produced to D. If `/_ACD = 112^@` and `/_B = 3/4 /_A`, then the mea sure of `/_B` is

Options

A. `30^@`
B. `48^@`
C. `45^@`
D. `64^@`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We are given:
- Angle ACD = 112°
- Angle B = (3/4) * Angle A

### Step 2: Use the exterior angle theorem
According to the exterior angle theorem, the exterior angle (ACD) is equal to the sum of the two opposite interior angles (A and B). Therefore, we can write the equation:
\[ \angle A + \angle B = \angle ACD \]
Substituting the known value:
\[ \angle A + \angle B = 112° \]

### Step 3: Substitute the expression for angle B
We know that:
\[ \angle B = \frac{3}{4} \angle A \]
Substituting this into the equation from Step 2:
\[ \angle A + \frac{3}{4} \angle A = 112° \]

### Step 4: Combine like terms
Combining the angles, we have:
\[ \frac{4}{4} \angle A + \frac{3}{4} \angle A = 112° \]
This simplifies to:
\[ \frac{7}{4} \angle A = 112° \]

### Step 5: Solve for angle A
To find angle A, multiply both sides by 4:
\[ 7 \angle A = 448° \]
Now, divide by 7:
\[ \angle A = \frac{448°}{7} = 64° \]

### Step 6: Find angle B
Now that we have angle A, we can find angle B using the relation:
\[ \angle B = \frac{3}{4} \angle A \]
Substituting the value of angle A:
\[ \angle B = \frac{3}{4} \times 64° = 48° \]

### Conclusion
The measure of angle B is:
\[ \angle B = 48° \]

---