IVEN: trapezoid ABCD

EF are the midpoints of segment AB and segment CD,
PROVE: segment EF is parallel to segment BC is parallel to AD , segment EF= one-half (AD + BC)

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To prove that segment EF is parallel to segment BC and AD, and that segment EF is equal to one-half (AD + BC), we can use the properties of trapezoids and the concept of midpoints.

Let's break down the proof step by step:

Step 1: Recall the definition of a trapezoid. A trapezoid is a quadrilateral with one pair of parallel sides. In our case, we have trapezoid ABCD.

Step 2: Recognize that EF are the midpoints of segment AB and segment CD. This means that EF divides both AB and CD into two equal parts.

Step 3: Let's consider the midpoints EF of AB and CD. Since EF is a midpoint, we can conclude that EF divides both sides into equal halves. Therefore, AE = EB and CF = FD.

Step 4: Now, we want to prove that segment EF is parallel to segment BC. To do this, we can use the property of mid-segment of a trapezoid. The mid-segment of a trapezoid is a line segment connecting the midpoints of the legs of the trapezoid. In our case, EF is the mid-segment of trapezoid ABCD.

Step 5: According to the mid-segment theorem, the mid-segment of a trapezoid is parallel to both bases (the parallel sides of the trapezoid). Therefore, we can conclude that EF is parallel to both BC and AD.

Step 6: Now, to prove that EF = 1/2(AD + BC), we can use the property of parallel lines and the concept of corresponding angles.

Step 7: Draw a line through A parallel to BC. Since EF is parallel to BC, the line we just drew intersects EF at a point G.

Step 8: By the corresponding angles property, angles AED and GED are congruent (since they are alternate angles formed by the transversal EF intersecting the parallel lines AD and AG).

Step 9: Similarly, since EF is parallel to BC, we can conclude that angle GED is congruent to angle GCB.

Step 10: Now, let's consider the triangles GED and GCB. They share side GE and have congruent angles. By the Angle-Side-Angle (ASA) congruence criterion, we can conclude that triangles GED and GCB are congruent.

Step 11: Therefore, we can say that GD = GC and GE = GB.

Step 12: Since EF is the mid-segment of trapezoid ABCD, we can conclude that GE is equal to one-half of AD, and EF is equal to one-half of BC.

Step 13: Combining steps 11 and 12, we get the desired result: EF = 1/2(AD + BC).

Hence, we have proven that segment EF is parallel to segment BC and AD, and that segment EF is equal to one-half (AD + BC).