In quadrilateral ABCD, side AB is parallel to side `DC. IFangleA:angleD=1:2andangleC:angleB=4:5.`
(i) Calculate wach angle of the quadrilateral.
(ii) Assign special name to quadrilateral ABCD.

Answer

To solve the problem step by step, we will follow the information given in the question and the relationships between the angles.

### Step 1: Understand the Given Ratios
We are given the following ratios:
- Angle A : Angle D = 1 : 2
- Angle C : Angle B = 4 : 5

### Step 2: Assign Variables to Angles
Let:
- Angle A = x
- Angle D = 2x (since the ratio is 1:2)
- Angle C = 4y
- Angle B = 5y (since the ratio is 4:5)

### Step 3: Use the Property of Parallel Lines
Since AB is parallel to DC, we can use the property that the sum of the angles on the same side of the transversal is equal to 180 degrees:
- Angle A + Angle D = 180 degrees
- Angle B + Angle C = 180 degrees

### Step 4: Set Up the Equations
From the relationships established:
1. \( x + 2x = 180 \)
2. \( 5y + 4y = 180 \)

### Step 5: Solve for x
From the first equation:
\[ 3x = 180 \]
\[ x = \frac{180}{3} = 60 \text{ degrees} \]

### Step 6: Solve for y
From the second equation:
\[ 9y = 180 \]
\[ y = \frac{180}{9} = 20 \text{ degrees} \]

### Step 7: Calculate Each Angle
Now we can find the measures of all angles:
- Angle A = x = 60 degrees
- Angle D = 2x = 2 * 60 = 120 degrees
- Angle C = 4y = 4 * 20 = 80 degrees
- Angle B = 5y = 5 * 20 = 100 degrees

### Step 8: Verify the Sum of Angles
We can verify that the angles satisfy the properties of a quadrilateral:
- Angle A + Angle B + Angle C + Angle D = 60 + 100 + 80 + 120 = 360 degrees

### Step 9: Identify the Type of Quadrilateral
Since one pair of opposite sides (AB and DC) is parallel, quadrilateral ABCD is a **trapezium** (or trapezoid in some regions).

### Final Answers:
(i) The angles of quadrilateral ABCD are:
- Angle A = 60 degrees
- Angle B = 100 degrees
- Angle C = 80 degrees
- Angle D = 120 degrees

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

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