This document discusses different types of quadrilaterals: - Parallelograms have both pairs of opposite sides parallel. - Rhombuses have four equal sides. - Rectangles have four right angles. - Squares are both rhombuses and rectangles, with four equal sides and four right angles. Theorems are presented about the properties of rhombuses, rectangles, and relationships between diagonals and the shape of the quadrilateral. Examples are given of proving properties of rhombuses using definitions and properties of parallel lines. Homework problems are assigned.

1. 8.4 Special Quadrilaterals January 10, 2013
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
≅ ∴
80
80 100
3. For what value of x is the quadrilateral a parallelogram? 50
2x
x + 30
Bellwork
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2. 8.4 Special Quadrilaterals January 10, 2013
8.4 Special Quadrilaterals
Rhombus: Rectangle: Square:
w/ w/ w/
4 ≅ sides 4 right <s
4 sides ≅
& 4 right <s
Rhombus Corollary: A quad is a rhombus iff it has 4 ≅ sides.
Rectangle Corollary: A quad is a rectangle iff it has 4 right <s.
Square Corollary: A quad is a square iff it is a rhombus and
rectangle.
Parallelograms
Rhombuses
Rectangles
Squares
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3. 8.4 Special Quadrilaterals January 10, 2013
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4. 8.4 Special Quadrilaterals January 10, 2013
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5. 8.4 Special Quadrilaterals January 10, 2013
Theorem 8.11
A is a rhombus iff its diagonals are .
Theorem 8.12
A is a rhombus iff each diagonal bisects a pair of
opposite <s.
Theorem 8.13
A is a rectangle iff its diagonals are ≅.
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6. 8.4 Special Quadrilaterals January 10, 2013
Rewrite the theorem as a biconditional and prove it.
Theorem 8.11
A is a rhombus iff its diagonals are .
Conditional Statement
If the diagonals of a are , then it is a rhombus.
Converse
If the is a rhombus, then its diagonals are .
PROOF:
Questions to ask.
Given: ABCD is a ; AC BD What do I know about //ograms?
Prove: ABCD is a rhombus. What do I know about rhombuses?
A B
X
D C
Statements Reasons
1. ABCD is a //ogram 1. Given
2. AC BD 2. Given
3. <CXB & <DXC are right <s 3. Def of lines
4. <CXB ≅ <DXC 4. Def of right <s
5. AD ≅ BC & DC ≅ AB 5. Opp. sides of are ≅
6. DX ≅ BX 6. Diagonals bisect each other in
7. CX ≅ XC 7. Reflexive Prop.
8. BXC ≅ DXC 8. SAS
9. BC ≅ DC 9. CPCTC
10. AD DC & BC AB 10. Transitive Prop.
≅ ≅
11. ABCD is a rhombus 11. Def of rhombus
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7. 8.4 Special Quadrilaterals January 10, 2013
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HW pg. 537 #3­51 mult. of 3 7

8. 8.4 Special Quadrilaterals January 10, 2013
8.4
HW
pg. 537
#3­51 multiples of 3
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