1) A quadrilateral is a parallelogram if its diagonals bisect each other and both pairs of opposite sides are congruent. 2) For a quadrilateral to be a parallelogram, x must be such that the opposite sides are parallel. 3) Theorems are presented about the properties of rhombuses, rectangles, and squares as special types of parallelograms.

1. Tell how you know that the quadrilateral is a parallelogram. 1.The diagonals bisect each other. 2. Both pairs of opp <s ≅ ∴ opp sides //. Def of //ogram 3. For what value of x is the quadrilateral a parallelogram? 50 Bellwork 80 100 80 2x x + 30

2. 8.4 Special Quadrilaterals Rhombus: w/ 4 ≅ sides Rectangle: w/ 4 right <s Square: w/ 4 sides ≅ & 4 right <s Rhombus Corollary: A quad is a rhombus iff it has 4 ≅ sides. Re ctangle Corollary: A quad is a rectangle iff it has 4 right <s. Sq uare Corollary: A quad is a square iff it is a rhombus and rectangle. Parallelograms Rhombuses Squares Rectangles

5. A is a rhombus iff its diagonals are . A is a rhombus iff each diagonal bisects a pair of opposite <s. A is a rectangle iff its diagonals are ≅ . Theorem 8.11 Theorem 8.12 Theorem 8.13

6. Theorem 8.11 Rewrite the theorem as a biconditional and prove it. Conditional Statement Converse PROOF: Statements Reasons ABCD is a //ogram Given Given <CXB & <DXC are right <s <CXB ≅ <DXC Def of right <s Questions to ask. What do I know about //ograms? What do I know about rhombuses? Reflexive Prop. SAS 6. CPCTC Transitive Prop. Def of rhombus ABCD is a rhombus A is a rhombus iff its diagonals are . If the diagonals of a are , then it is a rhombus. If the is a rhombus, then its diagonals are . Given: ABCD is a ; AC BD P rove: ABCD is a rhombus. A C D B X AC BD Def of lines DX ≅ BX Diagonals bisect each other in CX ≅ XC BXC ≅ DXC 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 1. 2. 3. 4. 5. 7. 8. 9. 10. 11. BC ≅ DC AD ≅ BC & DC ≅ AB Opp. sides of are ≅ AD ≅ DC & BC ≅ AB

8. 8.4 HW pg . 537 #3 -51 multiples of 3