Question

One angle of a quadrilateral is 48°. The other three angles are in ratio 6 : 9 : 11. Find these angles. What type of quadrilateral is this?

Answer 1

Given:

• One angle of the quadrilateral is 48°.
• The other three angles are in the ratio 6 : 9 : 11.

Step 1: Let the three angles be (6x), (9x) and (11x).

Step 2: The sum of all angles in any quadrilateral is 360°. Therefore, [48 + 6x + 9x + 11x = 360] [48 + 26x = 360]

Step 3: Solve for (x): [26x = 360 – 48 = 312] [x = `(312)/(26)` = 12]

Step 4: Find the three angles: [6x = 6 × 12 = 72°] [9x = 9 × 12 = 108°] [11x = 11 × 12 = 132°]

Step 5: List all angles of the quadrilateral: [48°, 72°, 108°, 132°]

Step 6: Determine the type of quadrilateral:

• Since all angles are not equal and no pair of angles is necessarily supplementary (sum 180°), the quadrilateral is not a rectangle, square, or parallelogram.
• The angles are all less than 180° and their sum is 360°, so it is a simple quadrilateral.
• Because one angle is acute (48°), another is obtuse (132°) and the others are in between, this is a general irregular quadrilateral.

The angles of the quadrilateral are 48°, 72°, 108° and 132°. The quadrilateral is an irregular quadrilateral (no special type like rectangle or square).