# Prove that if midpoints of non-parallel sides of a trapezium are joined, this line is parallel to parallel sides of the trapezium?

For proof see below.

Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the same ratio.

Reverse is also true that is, if a line divides the two sides of a triangle in the same ratio, then it is parallel to the third side.

Now let us revert to the problem. Let

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To prove that the line joining the midpoints of the non-parallel sides of a trapezium is parallel to the parallel sides of the trapezium, we can use the concept of parallel lines and the properties of trapeziums.

Let's denote the trapezium ABCD, where AB and CD are the parallel sides, and AD and BC are the non-parallel sides. Let E and F be the midpoints of sides AD and BC, respectively.

Now, connect E and F to form line EF. We want to show that EF is parallel to AB and CD.

First, consider triangles ADE and BCF. By definition, E and F are midpoints of their respective sides, which means AE is equal to ED and BF is equal to FC. Additionally, AD is parallel to BC (as they are opposite sides of the trapezium).

Now, considering triangles ADE and BCF, we have:

- AE = ED
- BF = FC
- AD || BC (given)

By the midpoint theorem, the segment EF is parallel to AD and BC and EF = 1/2(AD) = 1/2(BC).

Now, since AD and BC are parallel to AB and CD respectively, and EF is parallel to AD and BC, EF must also be parallel to AB and CD.

Therefore, the line joining the midpoints of the non-parallel sides of a trapezium is parallel to the parallel sides of the trapezium.

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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 A, B, C, and D. Sides A and B have a length of #3 # and sides C and D have a length of # 4 #. If the angle between sides A and C is #(7 pi)/18 #, what is the area of the parallelogram?
- A parallelogram has sides with lengths of #16 # and #9 #. If the parallelogram's area is #18 #, what is the length of its longest diagonal?
- Two rhombuses have sides with lengths of #8 #. If one rhombus has a corner with an angle of #(7pi)/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 #18 #. If one corner of the parallelogram has an angle of #(5 pi)/12 # and the parallelogram's area is #120 #, how long are the other two sides?
- If a quadrilateral is both a rectangle and a rhombus, then is it a square?

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