At least two pairs of consecutive angles are congruent (equal in measure) in which of the following quadrilaterals?

Introduction / Context:
This is a conceptual question about special types of quadrilaterals and the relationships between their interior angles. It tests whether you know which quadrilaterals have equal (congruent) consecutive angles and how many such pairs exist in each type. Understanding angle properties is essential for reasoning about shapes in geometry.

Given Data / Assumptions:

• We are considering standard Euclidean geometry.
• The quadrilaterals in the options are: parallelogram, isosceles trapezium, rhombus, and kite.
• “Consecutive angles” means adjacent angles that share a common side.
• We are asked where at least two pairs of consecutive angles are congruent.

Concept / Approach:
We recall the typical angle properties of each quadrilateral type:

• Parallelogram: opposite angles are equal; consecutive angles are supplementary (sum to 180°), not necessarily equal.
• Isosceles trapezium: one pair of opposite sides is parallel and the non parallel sides are equal; base angles on each base are equal.
• Rhombus: all sides are equal; opposite angles are equal; consecutive angles are supplementary in general, not equal unless the rhombus is a square.
• Kite: two pairs of adjacent sides are equal; typically only one pair of opposite angles is equal.

We look for a quadrilateral in which there are clearly two distinct pairs of adjacent angles that are equal.

Step-by-Step Solution:
Step 1: Consider an isosceles trapezium. Let the parallel sides be AB and CD, with non parallel sides AD and BC equal. Step 2: In an isosceles trapezium, the base angles at each base are equal. This means angle A equals angle B (on the top base), and angle C equals angle D (on the bottom base). Step 3: Angle A and angle B are consecutive (sharing side AB) and congruent. Angle C and angle D are also consecutive (sharing side CD) and congruent. Step 4: Therefore, the isosceles trapezium has at least two distinct pairs of consecutive equal angles. Step 5: For a general parallelogram, consecutive angles are supplementary, not equal, unless it becomes a rectangle, which is not explicitly given here. Step 6: A general rhombus has all sides equal but usually has two acute and two obtuse angles; only opposite angles, not adjacent ones, are equal, except in the special case of a square. Step 7: A kite typically has only one pair of opposite equal angles at the vertices where unequal sides meet; the consecutive angles are not usually congruent in pairs.

Verification / Alternative check:
To visualise, draw an isosceles trapezium with AB parallel to CD and AD and BC as equal legs. By basic geometry or symmetry arguments, the trapezium is symmetric about a perpendicular passing through the midpoints of AB and CD. This symmetry ensures that the angles at the ends of each base are equal: angle A = angle B and angle C = angle D, giving two pairs of adjacent equal angles. No other listed quadrilateral guarantees two such pairs by definition.

Why Other Options Are Wrong:
In a general parallelogram, the only guaranteed equal angles are opposite ones; consecutive angles sum to 180° and are not equal unless all angles are 90°, which defines a rectangle or square, not a generic parallelogram. In a rhombus, opposite angles are equal, but consecutive angles differ unless it is a square. In a kite, only one pair of opposite angles is guaranteed equal; there is no requirement that two distinct pairs of consecutive angles be congruent. Hence these options do not consistently satisfy the condition in all cases of that quadrilateral type.

Common Pitfalls:
Many learners loosely associate “all sides equal” with “many equal angles” and may incorrectly pick a rhombus. Others confuse properties of rectangles and squares with those of general parallelograms. It is important to distinguish general definitions from special cases and to pay attention to the term “consecutive angles,” which refers to angles sharing a side, not opposite angles across the quadrilateral.

Final Answer:
At least two pairs of consecutive angles are congruent in an isosceles trapezium.