In `DeltaABC, /_A + /_B = 145^@` and `/_C + 2/_B = 180^@`. State which one of the following relations is true ?

Options

A. `CA = AB `
B. `CA gt AB`
C. `BC lt AB `
D. `CA gt AB `

Answer

To solve the problem, we need to analyze the given conditions about the angles in triangle ABC.

### Step-by-Step Solution:

1. **Given Information**:
- \( \angle A + \angle B = 145^\circ \) (Equation 1)
- \( \angle C + 2\angle B = 180^\circ \) (Equation 2)

2. **Express \( \angle A \) in terms of \( \angle B \)**:
From Equation 1:
\[ \angle A = 145^\circ - \angle B \]

3. **Express \( \angle C \) in terms of \( \angle B \)**:
From Equation 2:
\[ \angle C = 180^\circ - 2\angle B \]

4. **Use the angle sum property of a triangle**:
The sum of angles in a triangle is \( 180^\circ \):
\[ \angle A + \angle B + \angle C = 180^\circ \]
Substituting the expressions for \( \angle A \) and \( \angle C \):
\[ (145^\circ - \angle B) + \angle B + (180^\circ - 2\angle B) = 180^\circ \]

5. **Simplify the equation**:
Combine like terms:
\[ 145^\circ + 180^\circ - 2\angle B = 180^\circ \]
\[ 325^\circ - 2\angle B = 180^\circ \]

6. **Isolate \( \angle B \)**:
Rearranging gives:
\[ 325^\circ - 180^\circ = 2\angle B \]
\[ 145^\circ = 2\angle B \]
\[ \angle B = \frac{145^\circ}{2} = 72.5^\circ \]

7. **Find \( \angle A \)**:
Substitute \( \angle B \) back into the expression for \( \angle A \):
\[ \angle A = 145^\circ - 72.5^\circ = 72.5^\circ \]

8. **Find \( \angle C \)**:
Substitute \( \angle B \) back into the expression for \( \angle C \):
\[ \angle C = 180^\circ - 2(72.5^\circ) = 180^\circ - 145^\circ = 35^\circ \]

9. **Analyze the angles**:
We have:
- \( \angle A = 72.5^\circ \)
- \( \angle B = 72.5^\circ \)
- \( \angle C = 35^\circ \)

10. **Determine the relationship between the sides**:
Since \( \angle A = \angle B \), sides opposite these angles (BC and AC) are equal:
\[ BC = AC \]
Since \( \angle C \) is the smallest angle, the side opposite \( \angle C \) (AB) is the smallest side:
\[ AB < AC \quad \text{and} \quad AB < BC \]

### Conclusion:
The correct relation is:
- \( AC > AB \) and \( BC > AB \)