Session 5, Part C: The Midline Theorem

Session 5, Part C:
The Midline Theorem

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

The midline theorem claims that cutting along the midline of a triangle creates a segment that is parallel to the base and half as long. Does that seem reasonable?

To prove the midline cut works, you need to use some geometry facts that you may already have encountered. If not, take some time to consider why these statements are true. Note 5

Fact 1: Vertical angles (the angles opposite each other when two lines intersect) are congruent (they have the same measure).

Why: We can show why, for example, m1 = m3:
m1 + m2 = 180° and m2 + m3 = 180°, since in both cases the two angles together create a "straight angle." So m1 + m2 = m2 + m3 = 180°. Subtracting m2 from each part of the equation, we see that
m1 = m3 = 180° - m2.

Fact 2: If two triangles have two sides that are the same length, and the angle between those two sides has the same measure, then the two triangles are congruent. The two triangles must have the same size and shape, so all three sides have the same length, and all three angles have the same measure. This is known as SAS (side-angle-side) congruence.

The single tick indicates the two sides that are the same length.
The double tick indicates thetwo sides that are the same length.
The angle markings indicate that those two angles have the same measure.

Why: The easiest way to be convinced of the fact that the two triangles are congruent is to draw some triangles. Draw a segment 2 inches long and a segment 3 inches long, with a 60° angle between them. Is there more than one way to complete the triangle? Come up with other cases to try.

Fact 3: There are many equivalent definitions of "parallelogram":

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A quadrilateral with both pairs of opposite sides parallel

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A quadrilateral with both pairs of opposite angles congruent

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A quadrilateral with both pairs of opposite sides congruent

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A quadrilateral with one pair of opposite sides both congruent and parallel

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The last definition is the one that will come in handy here. Go through the steps of understanding a definition in Session 3 if you're not sure why it works.



You may want to print this page for use in the problems that follow.

Next > Part C (Continued): Proving the Midline Theorem

Session 5: Index | Notes | Solutions | Video