Other Quadrilaterals
PROPERTIES OF SPEC IAL PARALLELOGRAMS
A rectangle is a parallelogram
with four right angles.
A rhombus is a parallelogram
with four congruent sides.
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A square is a parallelogram with four congruent sides
and four right angles.
Trapezoid
Definition: A quadrilateral with exactly one pair of parallel sides.
The parallel sides are called bases and the non-parallel sides are
called legs.
Legs
Trapezoid
Bases
Kite
Definition: A quadrilateral that has two pairs of consecutive
Kite
 sides
PROPERTIES OF SPEC IAL PARALLELOGRAMS
The Polygon Family Tree shows the relationships among
the different types of quadrilaterals.
Each shape has the properties of every group that it
belongs to. For instance, a square is a rectangle, a
rhombus, and a parallelogram, so it has all of the
properties of each of those shapes.
Moving up the Polygon Family
Tree shows a relationship that
is always true.
Moving down the Polygon
Family Tree shows a
relationship that is sometimes
true.
***However, a figure below
holds ALL the properties of
the figures above it.
1) Describing a Special Parallelogram
Decide whether the statement is always, sometimes, or never true.
1. A rhombus is a rectangle.
The statement is sometimes true.
2. A parallelogram is a rectangle.
The statement is sometimes true.
3. A square is a rectangle.
The statement is always true.
USING DIAGONALS OF SPECIAL PARALLELOGRAMS
THEOREMS
THEOREM 6.11
A parallelogram is a rhombus if and only if its
diagonals are perpendicular.
ABCD is a rhombus if and only if AC
BD
What other quadrilateral would this then be true for?
Square
USING DIAGONALS OF SPECIAL PARALLELOGRAMS
THEOREMS
THEOREM 6.12
B
C
A parallelogram is a rhombus if and only if each
diagonal bisects a pair of opposite angles.
A
ABCD is a rhombus if and only if AC bisects
BD bisects
D
DAB and
ADC and
What other quadrilateral would this then be true for?
BCD and
CBA
Square
USING DIAGONALS OF SPECIAL PARALLELOGRAMS
THEOREM S
THEOREM 6.13
A
B
D
C
A parallelogram is a rectangle if and
only if its diagonals are congruent.
ABCD is a rectangle if and only if AC  BD
What other quadrilateral would this then be true for?
Square
Properties of Kites
ABCD is a kite if and only if 𝐴𝐶 ⊥ 𝐵𝐷
2) Using Properties of Parallelograms
Decide if the statement is always, sometimes, or never true (A, S or N).
1) If the diagonals bisect each other, then it is a rhombus.
Sometimes
2) If a quadrilateral is a square, then its diagonals are perpendicular.
Always
3) If the diagonals bisect the angles, then it is a square.
Sometimes
4) If both pairs of opposite sides are parallel, then it is a rectangle.
Sometimes
3) Using Properties of Parallelograms
What quadrilaterals do the following descriptions fit?
1) All sides congruent
Rhombus/Square
2) All angles congruent
Rectangle/Square
3) Diagonals congruent
Rectangle/Square
4) Opposite angles congruent
Parallelogram, Square,
Rectangle, Rhombus
4) Using Properties of a Rhombus
In the diagram, PQRS is a rhombus.
What is the value of x?
P
Q
3x + 4
S
SOLUTION
All four sides of a rhombus are congruent, so RS = PS.
3x + 4 = 5x - 6
4 = 2x - 6
10 = 2x
x=5
5x - 6
R
Ex 5) 25 from HW!
Prove: ∆ART ≅ ∆𝐴𝐶𝐸
1.
RECT is a rectangle
2. RECT is a parallelogram
Given
All rectangles are parallelograms
 Diagonals bisect
3. 𝑅𝐴 ≅ 𝐶𝐴, 𝑇𝐴 ≅ 𝐸𝐴
4. ∠RAT ≅ ∠CAE
Vertical angles ≅
5. ∆ART ≅ ∆𝐴𝐶𝐸
SAS