Midsegment of a trapezoid

We can demonstrate that there is a midsegment in a trapezoid if at least one of the following conditions is met:

1. If in a trapezoid there is a straight line that extends from the midpoint of one side to the midpoint of another side, we can determine that it is a midsegment and, therefore, it is parallel to both bases of the trapezoid and measures half the length of these.
2. If in a trapezoid there is a straight line that comes out of one side and is parallel to one of the bases of the trapezoid, we can determine that it is a midsegment and, therefore, it is parallel to both bases of the trapezoid, measures half the length of these two and also cuts the second side it touches in half.

Data:

$AE=DE$
$AB‖EF$

To prove:
$BF=FC$

Solution:
According to the second rule of the theorem,
if in the trapezoid, a straight line $EF$ comes out from the midpoint of one of the sides - we know that: $AE=DE$
and is also parallel to one of the bases of the trapezoid – Given that: $AB‖EF$
we can determine that it is the mid-segment of the trapezoid.
Consequently, we can determine that it also cuts through the midpoint of the other side it touches.
That is to say:
$BF=FC$

If you are interested in this article, you might also be interested in the following articles:

• Midsegment
• Midsegment of a triangle
• Sum of the interior angles of a polygon
• Angles in regular hexagons and octagons
• Measure of an angle of a regular polygon
• Sum of the exterior angles of a polygon
• Relationships between angles and sides of a triangle
• The relationship between the lengths of the sides of a triangle
• Identification of an isosceles triangle
• How is the area of a trapezoid calculated?
• Symmetry in trapezoids
• Diagonals of an isosceles trapezoid
• How is the perimeter of a trapezoid calculated?
• Characteristics and types of trapezoids

In the Tutorela blog, you will find a variety of articles on mathematics.