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Difficulty: MediumProperties of Quadrilaterals

In any parallelogram ABCDABCD, if consecutive angles ∠A\angle A and ∠B\angle B are supplementary, then quadrilateral ABCDABCD must be a rectangle.

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Answer

False. Consecutive interior angles are supplementary in all parallelograms, not only in rectangles.
The statement is false because consecutive interior angles are supplementary in every parallelogram due to parallel opposite sides. This property does not imply that the angles are right angles (90∘90^\circ); for instance, a parallelogram with angles measuring 60∘60^\circ and 120∘120^\circ has supplementary consecutive angles but is clearly not a rectangle.

Step-by-Step Solution

1
Recall the consecutive angle property for any general parallelogram.
In any parallelogram ABCDABCD, opposite sides are parallel (AD∥BCAD \parallel BC). When parallel lines are intersected by a transversal ABAB, consecutive interior angles are supplementary: ∠A+∠B=180∘\angle A + \angle B = 180^\circ.
Consecutive interior angles formed by parallel lines and a transversal always sum to 180∘180^\circ.
2
Compare this general property with the specific condition that defines a rectangle.
A parallelogram is a rectangle if and only if all four interior angles are right angles (90∘90^\circ), which requires consecutive angles to be congruent (equal in measure), not merely supplementary.
Two supplementary angles can have measures such as 60∘60^\circ and 120∘120^\circ, which form an oblique parallelogram rather than a rectangle.
3
Determine the truth value of the statement.
Because the condition of having supplementary consecutive angles is satisfied by every parallelogram and does not guarantee 90∘90^\circ angles, the statement is false.
A property shared by all members of a general class (parallelograms) cannot be used as a sufficient condition to classify a figure into a restrictive subclass (rectangles).

Key Concept

Properties of Quadrilaterals: Parallelogram vs. Rectangle Angle Rules
Estimated Time:1m 0s
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Topics

Properties of Quadrilaterals

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