In a proof that a quadrilateral is a parallelogram, show that one of the five conditions is satisfied.
Example ①
Let M and N be the midpoints of sides AB and BC of △ABC, respectively. Show that quadrilateral AMBC, formed by combining △AMN and △CBN, is a parallelogram.
(Hint: Use the midpoint theorem.)
Solution (Introduction to the Midpoint Theorem): In △ABC, if M and N are the midpoints of AB and BC, respectively, then MN∥AC and MN=AC/2.
Let M and N be the midpoints of sides AB and BC of △ABC, respectively. Show that quadrilateral AMBC, formed by combining △AMN and △CBN, is a parallelogram.
(Hint: Use the midpoint theorem.)
Solution (Introduction to the Midpoint Theorem): In △ABC, if M and N are the midpoints of AB and BC, respectively, then MN∥AC and MN=AC/2.
Key Point
- State the number of the “condition for being a parallelogram” as the reason.
- Midpoint theorem: The segment joining the midpoints of two sides of a triangle is parallel to and half the length of the third side.
Practice Problem
- When E, F, G, and H are the midpoints of the sides of quadrilateral ABCD, state that EFGH is a parallelogram (using the midpoint theorem).
Answer and Explanation
- Answer: With respect to diagonal AC, EF∥AC and EF=AC/2, and HG∥AC and HG=AC/2. Therefore, EF∥HG and EF=HG (condition ⑤). EFGH is a parallelogram.