The measures of the angles of a quadrilateral taken in order are proportional to `1:2:3:4`, then the quadrilateral is :

Options

A. parallelogram
B. trapezium
C. rectangle
D. rhombus

Answer

To solve the problem of finding the type of quadrilateral based on the proportional measures of its angles, we can follow these steps:

### Step 1: Define the Angles
Let the angles of the quadrilateral be represented as follows based on the given ratio of 1:2:3:4:
- First angle = \( x \)
- Second angle = \( 2x \)
- Third angle = \( 3x \)
- Fourth angle = \( 4x \)

### Step 2: Set Up the Equation
We know that the sum of the angles in a quadrilateral is always 360 degrees. Therefore, we can set up the equation:
\[ x + 2x + 3x + 4x = 360 \]

### Step 3: Simplify the Equation
Combine the terms on the left side:
\[ 10x = 360 \]

### Step 4: Solve for \( x \)
To find the value of \( x \), divide both sides of the equation by 10:
\[ x = \frac{360}{10} = 36 \]

### Step 5: Calculate Each Angle
Now that we have the value of \( x \), we can find the measures of each angle:
- First angle = \( x = 36^\circ \)
- Second angle = \( 2x = 2 \times 36 = 72^\circ \)
- Third angle = \( 3x = 3 \times 36 = 108^\circ \)
- Fourth angle = \( 4x = 4 \times 36 = 144^\circ \)

### Step 6: Identify the Type of Quadrilateral
Now we need to determine what type of quadrilateral these angles correspond to:
- A **parallelogram** has opposite angles that are equal.
- A **rectangle** has all angles equal to \( 90^\circ \).
- A **rhombus** also has opposite angles equal.
- A **trapezium** (or trapezoid) can have different angles.

Since we have angles of \( 36^\circ, 72^\circ, 108^\circ, \) and \( 144^\circ \), none of these angles are equal, and they do not satisfy the properties of a parallelogram, rectangle, or rhombus. Therefore, the quadrilateral must be a **trapezium**.

### Final Answer
The quadrilateral is a **trapezium**.

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