1. Math 2 Geometry Based on Elementary Geometry, 3 rd ed, by Alexander & Koeberlein 4.2 The Parallelogram and Kite

2. One of the objectives of this section is to determine under what conditions is a quadrilateral a parallelogram.

3. Recall the definition of a parallelogram: a quadrilateral in which both pairs of opposite sides are parallel.

4. Theorem 4.2.1 If two sides of a quadrilateral are both congruent and parallel, then, the quadrilateral is a parallelogram.

5. Theorem 4.2.1 If two sides of a quadrilateral are both congruent and parallel, then, the quadrilateral is a parallelogram. Proof?

6. Theorem 4.2.1 If two sides of a quadrilateral are both congruent and parallel, then, the quadrilateral is a parallelogram. Proof? We need drawing Given statements Prove statements

7. Theorem 4.2.2 If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.

8. Theorem 4.2.3 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

9. Definition A kite is a quadrilateral with two distinct pairs of congruent adjacent sides.

10. Definition A kite is a quadrilateral with two distinct pairs of congruent adjacent sides.

11. Definition A kite is a quadrilateral with two distinct pairs of congruent adjacent sides.

12. Theorem 4.2.4 In a kite, one pair of opposite angles are congruent. Which angles?

13. We can use the properties and theorems of parallelograms to get more insight into triangles…

14. Theorem 4.2.5 The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one- half the length of the third side.

15. Theorem 4.2.5 The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one- half the length of the third side.

16. Theorem 4.2.5 The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one- half the length of the third side.

17. Theorem 4.2.5 The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one- half the length of the third side. Proof?

18. Theorem 4.2.5 The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one- half the length of the third side.

19. Theorem 4.2.5 The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one- half the length of the third side. First prove segment and third side are ||

20. Theorem 4.2.5 The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one- half the length of the third side. First prove segment and third side are || Parallelogram

21. Theorem 4.2.5 The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one- half the length of the third side. Next, prove length of the segment Is half the length of the third side

22. Theorem 4.2.5 The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one- half the length of the third side. Next, prove length of the segment Is half the length of the third side   s

23. Example Given: ∆ ABC as shown with D the midpoint of seg AC and E the midpoint of seg BC; DE = 2x + 1 AB = 5x – 1 Find: x, DE, and AB What theorem allows us to write what equation? C DE AB