# How do I prove that if a quadrilateral has consecutive angles that are supplementary, then it is a parallelogram?

Not necessary

Consider a trapezoid with angles 35-135-90-90. It has 2 instances of consecutive (adjacent?) angles that are supplementary (35-135 and 90-90) but it is not a parallelogram

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To prove that if a quadrilateral has consecutive angles that are supplementary, then it is a parallelogram, you can use the following approach:

- Start by labeling the quadrilateral vertices as ( A), ( B), ( C), and ( D), in clockwise or counterclockwise order.
- Assume that consecutive angles are supplementary. That is, angle ( A) + angle ( B) = 180 degrees, angle ( B) + angle ( C) = 180 degrees, angle ( C) + angle ( D) = 180 degrees, and angle ( D) + angle ( A) = 180 degrees.
- Consider the properties of supplementary angles to prove that opposite angles are equal. Since angle ( A) + angle ( B) = 180 degrees, and angle ( C) + angle ( D) = 180 degrees, then angle ( A) = angle ( C) and angle ( B) = angle ( D).
- Use the fact that opposite angles are equal to show that opposite sides are parallel. This follows from the properties of angles formed when a transversal intersects parallel lines.
- Therefore, if opposite sides of a quadrilateral are parallel, it is a parallelogram.

By demonstrating these steps, you establish that a quadrilateral with consecutive supplementary angles must be a parallelogram.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- A parallelogram has sides with lengths of #7 # and #16 #. If the parallelogram's area is #14 #, what is the length of its longest diagonal?
- Two opposite sides of a parallelogram each have a length of #6 #. If one corner of the parallelogram has an angle of #(3pi)/8 # and the parallelogram's area is #48 #, how long are the other two sides?
- Two rhombuses have sides with lengths of #6 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #pi/6 #, what is the difference between the areas of the rhombuses?
- Two opposite sides of a parallelogram each have a length of #9 #. If one corner of the parallelogram has an angle of #(3pi)/8 # and the parallelogram's area is #26 #, how long are the other two sides?
- What quadrilateral has no congruent sides?

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