In `/_\ABC`, if `/_A =70^(@), /_B=80^(@)" and "/_C=30^(@)`, then which of the following is the correct relation among the sides of the triangle?
`AB lt BC lt CA`
`AB gt BC gt CA`
`BC lt AB lt CA`
`AC lt AB lt BC`

Options

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

Answer

To solve the problem, we need to determine the relationship between the sides of triangle ABC based on the given angles. Here are the steps to find the correct relation among the sides:

### Step 1: Identify the angles
We are given the angles of triangle ABC:
- Angle A = 70°
- Angle B = 80°
- Angle C = 30°

### Step 2: Understand the relationship between angles and sides
In a triangle, the side opposite the largest angle is the longest side, and the side opposite the smallest angle is the shortest side. Therefore, we need to compare the angles to determine the sides.

### Step 3: Compare the angles
- Angle B (80°) is the largest angle.
- Angle A (70°) is the second largest angle.
- Angle C (30°) is the smallest angle.

### Step 4: Determine the sides based on the angles
- The side opposite Angle B (80°) is AC. Therefore, AC is the longest side.
- The side opposite Angle A (70°) is BC. Therefore, BC is the second longest side.
- The side opposite Angle C (30°) is AB. Therefore, AB is the shortest side.

### Step 5: Write the relationship among the sides
From the above comparisons, we can conclude:
- AB < BC < AC

### Final Answer
Thus, the correct relation among the sides of triangle ABC is:
**AB < BC < AC**