In a `Delta ABC, angle A+ angle B =145^(@), angle C+2angleB =180^(@)` find the relation b/w AB & AC.
(i) `AC gt AB`
(ii) `AC=AB`
(iii) `AC lt AB`

Answer

To find the relation between sides AB and AC in triangle ABC given the conditions on angles A, B, and C, we can follow these steps:

### Step 1: Set up the equations based on the given information.
We are given:
1. \( \angle A + \angle B = 145^\circ \) (Equation 1)
2. \( \angle C + 2\angle B = 180^\circ \) (Equation 2)

### Step 2: Express angle C in terms of angle B.
From Equation 2:
\[ \angle C = 180^\circ - 2\angle B \]

### Step 3: Substitute angle C into the triangle angle sum property.
The sum of angles in a triangle is \( 180^\circ \):
\[ \angle A + \angle B + \angle C = 180^\circ \]
Substituting for \( \angle C \):
\[ \angle A + \angle B + (180^\circ - 2\angle B) = 180^\circ \]
This simplifies to:
\[ \angle A - \angle B = 0 \]
Thus, we can express angle A in terms of angle B:
\[ \angle A = \angle B \]

### Step 4: Substitute angle A back into Equation 1.
Using \( \angle A = \angle B \) in Equation 1:
\[ \angle B + \angle B = 145^\circ \]
This simplifies to:
\[ 2\angle B = 145^\circ \]
Thus:
\[ \angle B = 72.5^\circ \]

### Step 5: Find angle A and angle C.
Since \( \angle A = \angle B \):
\[ \angle A = 72.5^\circ \]
Now substitute \( \angle B \) back into the expression for \( \angle C \):
\[ \angle C = 180^\circ - 2(72.5^\circ) = 180^\circ - 145^\circ = 35^\circ \]

### Step 6: Analyze the angles to find the relation between sides AB and AC.
In triangle ABC:
- \( \angle A = 72.5^\circ \)
- \( \angle B = 72.5^\circ \)
- \( \angle C = 35^\circ \)

According to the properties of triangles, the side opposite the largest angle is the longest. Here, angles A and B are equal, and they are larger than angle C. Therefore, sides opposite angles A and B (which are AB and AC respectively) are equal.

### Conclusion:
Since \( \angle A = \angle B \), we conclude:
\[ AC = AB \]