The measures of the four successive interior angles of a quadrilateral are in the ratio 7 : 11 : 7 : 11. Which type of quadrilateral does this describe?

Introduction / Context:
This question asks you to classify a quadrilateral based on the ratio of its successive interior angles. It tests basic angle sum facts for quadrilaterals and recognition of angle patterns that characterise special quadrilaterals such as parallelograms, rectangles, squares and trapezia. Understanding angle relationships is an important part of elementary geometry.

Given Data / Assumptions:

• Four successive angles of a quadrilateral are in the ratio 7 : 11 : 7 : 11.
• Let the common multiplying factor be x degrees.
• The sum of interior angles of any quadrilateral is 360 degrees.
• We need to see what type of quadrilateral this angle set corresponds to.
• All angles are assumed to be positive and the quadrilateral is simple (non self intersecting).

Concept / Approach:
If the angles are in the ratio 7 : 11 : 7 : 11, then the actual angles are 7x, 11x, 7x and 11x. The sum of these must be 360 degrees. Solving for x gives the specific angle values. If opposite angles are equal and each pair of adjacent angles are supplementary (sum to 180 degrees), then the quadrilateral is a parallelogram. Rectangles and squares require all angles to be 90 degrees, and a trapezium only has one pair of parallel sides and does not necessarily have equal opposite angles.

Step-by-Step Solution:
Step 1: Let the four angles be 7x, 11x, 7x and 11x degrees. Step 2: Sum of interior angles of a quadrilateral is 360 degrees, so 7x + 11x + 7x + 11x = 360. Step 3: Combine terms: (7 + 11 + 7 + 11)x = 36x = 360. Step 4: Solve for x: x = 360 / 36 = 10 degrees. Step 5: Hence the four angles are 70°, 110°, 70° and 110°. Step 6: Opposite angles are equal (70° opposite 70°, 110° opposite 110°), and adjacent angles are supplementary since 70° + 110° = 180°. Step 7: A quadrilateral with equal opposite angles and pairs of adjacent angles that sum to 180° is a parallelogram.

Verification / Alternative check:
In a parallelogram, each pair of opposite angles is equal and each pair of adjacent angles is supplementary. The computed angles 70°, 110°, 70° and 110° satisfy both conditions. A rectangle or square would require all four angles to be 90°, which is not the case here. A kite has equal adjacent sides and does not have the given angle structure. A single pair of parallel sides, as in a trapezium, does not force both pairs of opposite angles to be equal. Therefore the only consistent classification is parallelogram.

Why Other Options Are Wrong:
A trapezium does not generally have equal opposite angles. A rectangle and a square both have all angles equal to 90°, which does not match 70° and 110°. A kite has a different angle pattern and often only two equal angles. None of these match the computed set of angles except a general parallelogram. Hence the options trapezium, rectangle, square and kite are not correct classifications for this angle ratio.

Common Pitfalls:
Some learners see two distinct angle measures and may think of a kite or a trapezium without checking angle sums and opposite angle equality. Another mistake is forgetting that not only the sum of angles matters, but also how they are arranged. Always compute the actual angles, then verify which quadrilateral properties they satisfy instead of guessing based only on ratios.

Final Answer:
The quadrilateral is a parallelogram.