`alpha,beta,gammaand delta` are angles in I,II,II and IV quadrants, respectively and none of them is an integral multiple of `pi//2`. They form an increasing arithmetic progression.
if `alpha+beta+gamma+delta=thetaand alpha70^@`, then

Options

A. `400^@ltthetalt580^@`
B. `470^@ltthetalt650^@`
C. `680^@ltthetalt860^@`
D. `540^@ltthetalt900^@`

Explanation

To solve the problem step by step, we need to analyze the information given about the angles α, β, γ, and δ, which are in the first, second, third, and fourth quadrants respectively, and form an increasing arithmetic progression.

### Step 1: Define the angles in terms of α and the common difference d
Since α, β, γ, and δ are in an arithmetic progression, we can express them as:
- α = α (given as 70°)
- β = α + d
- γ = α + 2d
- δ = α + 3d

### Step 2: Determine the ranges of the angles
- α is in the first quadrant: 0° < α < 90° (here α = 70°)
- β is in the second quadrant: 90° < β < 180°
- γ is in the third quadrant: 180° < γ < 270°
- δ is in the fourth quadrant: 270° < δ < 360°

### Step 3: Substitute α into the expressions for β, γ, and δ
Substituting α = 70° into the expressions for β, γ, and δ:
- β = 70° + d
- γ = 70° + 2d
- δ = 70° + 3d

### Step 4: Establish inequalities based on the quadrant conditions
From the conditions of the quadrants, we can set up the following inequalities:
1. For β:
\[ 90° < 70° + d < 180° \]
This simplifies to:
\[ 20° < d < 110° \]

2. For γ:
\[ 180° < 70° + 2d < 270° \]
This simplifies to:
\[ 55° < d < 100° \]

3. For δ:
\[ 270° < 70° + 3d < 360° \]
This simplifies to:
\[ 66.67° < d < 96.67° \]

### Step 5: Find the common range for d
From the inequalities derived:
- From β: \(20° < d < 110°\)
- From γ: \(55° < d < 100°\)
- From δ: \(66.67° < d < 96.67°\)

The common range for d is:
\[ 66.67° < d < 96.67° \]

### Step 6: Calculate θ
Now, we know that:
\[ \theta = α + β + γ + δ \]
Substituting the expressions:
\[ \theta = α + (α + d) + (α + 2d) + (α + 3d) = 4α + 6d \]
Substituting α = 70°:
\[ \theta = 4(70°) + 6d = 280° + 6d \]

### Step 7: Determine the range for θ
Using the range for d:
- Minimum value of d (66.67°):
\[ \theta_{min} = 280° + 6(66.67°) = 280° + 400° = 680° \]
- Maximum value of d (96.67°):
\[ \theta_{max} = 280° + 6(96.67°) = 280° + 580° = 860° \]

Thus, the range for θ is:
\[ 680° < θ < 860° \]

### Conclusion
The value of θ lies between 680° and 860°.