Session 5, Part C:
The Midline Theorem

In This Part: The Midline Cut | Some Geometry Facts | Proving the Midline Theorem

 With these facts in mind, let's prove that the midline cut works. Cutting along the midline of a triangle creates a segment that is parallel to the base and half as long. Start with this picture. It was created so that D and E are midpoints of AC and BC respectively. D, E, and F are collinear (on the same line), and DE and EF are the same length. Note 6 Problem C1 Considering the facts on the previous page, explain why triangles DEC and FEB are congruent.

 Problem C2 Explain why AD and BF have the same length.

 Problem C3 Explain why AD and BF are parallel.

 Look at the angles marked 1 and 2 in the picture below.   Close Tip Look at the angles marked 1 and 2 in the picture below.

 Problem C4 Explain why ABFD is a parallelogram.

 Problem C5 Explain why DE is parallel to AB and half as long.

By proving the midline theorem -- since we proved it, we can call it a theorem now -- we illustrate some important mathematical ideas.

 • First, you know that the midline cut will always work, on any triangle. There is nothing special about triangle ABC in the picture. So now, no matter how long and skinny, or how funny-looking a triangle you get, you can be sure the midline theorem will hold. You could never test it out on every triangle because there is an infinite number of cases to try, so a proof is the only way to be sure. • Second, now that you're sure the midline theorem will holds, you can use it when you look at other problems.

 Video Segment Being able to make these kinds of proofs can be a useful skill in the professional world. In this segment, Mr. Ialeggio explains how, in order to ensure accurate dimensions, he proves that the window frames he makes for antique homes are indeed rectangles. If you are using a VCR, you can find this segment on the session video approximately 23 minutes and 29 seconds after the Annenberg Media logo.

 Problem C6 In the picture below, AB equals BC, and AD equals CD. Explain why EF is parallel to HG and why they are the same length. What kind of quadrilateral is EFGH?

Consider the diagonal AC and its relationship to the segments in question.   Close Tip

 Problem C7 Draw another quadrilateral and connect the midpoints of the four sides in order. What kind of quadrilateral is formed and why? Note 7

 Problem C1-C6 adapted from Connected Geometry, developed by Educational Development Center, Inc. pp.194-195. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math

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 Session 5: Index | Notes | Solutions | Video