In quadrilateral ABCD, side AB is parallel to side DC. If `angleA : angle D=1:2 and angleC: angleB=4:5`.
(i) Calculate each angle of the quadrilateral.
(ii) Assign a special name to quadrialateral ABCD.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationships between the angles
We know that in quadrilateral ABCD, side AB is parallel to side DC. Therefore, angles A and D are co-interior angles, and angles B and C are also co-interior angles. This gives us the following equations:
- \( \angle A + \angle D = 180^\circ \)
- \( \angle B + \angle C = 180^\circ \)

### Step 2: Set up the ratios for angles A and D
We are given that the ratio of angle A to angle D is 1:2. We can express this as:
- Let \( \angle A = x \)
- Then \( \angle D = 2x \)

### Step 3: Substitute into the equation for angles A and D
Using the equation from Step 1:
\[ \angle A + \angle D = 180^\circ \]
Substituting the expressions for angles A and D:
\[ x + 2x = 180^\circ \]
This simplifies to:
\[ 3x = 180^\circ \]

### Step 4: Solve for angle A
Now, divide both sides by 3:
\[ x = \frac{180^\circ}{3} = 60^\circ \]
Thus:
- \( \angle A = 60^\circ \)
- \( \angle D = 2x = 2 \times 60^\circ = 120^\circ \)

### Step 5: Set up the ratios for angles B and C
Next, we are given that the ratio of angle C to angle B is 4:5. We can express this as:
- Let \( \angle C = 4y \)
- Then \( \angle B = 5y \)

### Step 6: Substitute into the equation for angles B and C
Using the equation from Step 1:
\[ \angle B + \angle C = 180^\circ \]
Substituting the expressions for angles B and C:
\[ 5y + 4y = 180^\circ \]
This simplifies to:
\[ 9y = 180^\circ \]

### Step 7: Solve for angle B
Now, divide both sides by 9:
\[ y = \frac{180^\circ}{9} = 20^\circ \]
Thus:
- \( \angle B = 5y = 5 \times 20^\circ = 100^\circ \)
- \( \angle C = 4y = 4 \times 20^\circ = 80^\circ \)

### Step 8: Summarize the angles
Now we have calculated all the angles:
- \( \angle A = 60^\circ \)
- \( \angle B = 100^\circ \)
- \( \angle C = 80^\circ \)
- \( \angle D = 120^\circ \)

### Step 9: Identify the type of quadrilateral
Since AB is parallel to DC and the other sides are not parallel, quadrilateral ABCD is a **trapezium**.

### Final Answer
(i) The angles of quadrilateral ABCD are:
- \( \angle A = 60^\circ \)
- \( \angle B = 100^\circ \)
- \( \angle C = 80^\circ \)
- \( \angle D = 120^\circ \)

(ii) Quadrilateral ABCD is a **trapezium**.