Bellwork

1. Clickers Bellwork A B • What congruence postulate shows that triangle ABE is congruent to triangle CDE • If angle E is congruent to angle G, find the measure of angle E • What is the probability of pulling a 7 of hearts from a standard deck of cards E D C G H 62o 62o F E

2. Bellwork Solution • What congruence postulate shows that triangle ABE is congruent to triangle CDE A B E D C

3. Bellwork Solution • If angle E is congruent to angle G, find the measure of angle E G H 62o 62o F E

4. Bellwork Solution • What is the probability of pulling a 7 of hearts from a standard deck of cards

5. Section 8.3 Show that a Quadrilateral is a Parallelogram

6. The Concept • Thus far we’ve been talking about using the properties of parallelograms • Today we’re going to reverse those properties and use them to show that an object is a parallelogram • Our methodology is somewhat similar to reversing our angle relationships in parallel lines to prove parallelility

7. Theorems Theorem 8.7: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram Theorem 8.8: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram Aren’t these just the converses of Theorems 8.3, 8.4, 8.5 & 8.6?

8. Example Let’s look at Example #1 in the book on page 523. Axis of symmetry Vertex

9. Theorems Theorem 8.9: If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram Theorem 8.10: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram Set of Parallel Sides

10. Example What theorem would we use to show that the quadrilateral is a parallelogram? Axis of symmetry Vertex

11. Example What theorem would we use to show that the quadrilateral is a parallelogram? Axis of symmetry Vertex

12. Example What theorem would we use to show that the quadrilateral is a parallelogram? Axis of symmetry Vertex

13. Example What theorem would we use to show that the quadrilateral is a parallelogram? Axis of symmetry Vertex

14. Example What theorem would we use to show that the quadrilateral is a parallelogram? Axis of symmetry Vertex

15. Example What value of x, makes the quadrilateral a parallelogram? Axis of symmetry Vertex

16. Proving via Coordinates We can also prove that an object is a quadrilateral if we’re given the coordinates of the vertices by • Proving that both sets of sides are congruent • Proving that two sides are congruent and parallel • Proving that both sets of sides are parallel Proving congruency is easy via the distance formula, so let’s look at this one

17. Proving via Coordinates Prove that a quadrilateral with vertices A(-5,0), B(0,4), C(3,0), D(-2,-4) is a parallelogram Proving parallelism is done by looking at the slopes of the two lines and determining if they are equal

18. Homework • 8.3 • 1-11, 17-21, 27, 31, 32

19. Example Which of the following is a parallelogram Axis of symmetry Vertex

20. Most Important Points • Converses of Theorems • Method for proving parallelogram in coordinate plane