If the angles `A, B, C, D` of a quadrilateral, taken in order are in the ratio `7 : 13 : 12 : 8`, then `ABCD` is:

Options

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

Answer

To solve the problem, we need to find the angles of the quadrilateral ABCD given that the angles are in the ratio 7:13:12:8. We will follow these steps:

### Step-by-Step Solution:

1. **Understand the Ratio**: The angles A, B, C, and D are in the ratio 7:13:12:8. This means we can express the angles in terms of a variable \( x \):
- Angle A = \( 7x \)
- Angle B = \( 13x \)
- Angle C = \( 12x \)
- Angle D = \( 8x \)

2. **Sum of Angles in a Quadrilateral**: The sum of the angles in any quadrilateral is always 360 degrees. Therefore, we can set up the equation:
\[ 7x + 13x + 12x + 8x = 360 \]

3. **Combine Like Terms**: Combine the terms on the left side of the equation:
\[ (7 + 13 + 12 + 8)x = 360 \]
\[ 40x = 360 \]

4. **Solve for \( x \)**: Divide both sides of the equation by 40 to find the value of \( x \):
\[ x = \frac{360}{40} = 9 \]

5. **Calculate Each Angle**: Now that we have the value of \( x \), we can find each angle:
- Angle A = \( 7x = 7 \times 9 = 63^\circ \)
- Angle B = \( 13x = 13 \times 9 = 117^\circ \)
- Angle C = \( 12x = 12 \times 9 = 108^\circ \)
- Angle D = \( 8x = 8 \times 9 = 72^\circ \)

6. **Determine the Type of Quadrilateral**: Now we need to determine what type of quadrilateral ABCD is based on the angles:
- Since all angles are distinct (63°, 117°, 108°, and 72°), ABCD cannot be a parallelogram or a rhombus (where opposite angles are equal).
- Check if it is a trapezium: For a trapezium, at least one pair of opposite angles must be supplementary (sum to 180°).
- Angle A + Angle B = 63° + 117° = 180° (this is one pair)
- Angle C + Angle D = 108° + 72° = 180° (this is another pair)
- Since both pairs of angles are supplementary, ABCD is a trapezium.

### Final Answer:
The quadrilateral ABCD is a trapezium.