Download Chapter 4 Congruent Triangles (page 116)

Lesson 5-4:
Special Quadrilaterals
(page 184)
Essential Question
How can the properties of quadrilaterals
be used to solve real life problems?
RECTANGLE: a quadrilateral with
four right angles.
Every rectangle is a _____;
Both pairs of opposite ∠’s are ≅
reason:____________________________________________.
Theorem 5-12
The diagonals of a rectangle are
congruent .
A
B
D
C
Theorem 5-16
If an angle of a parallelogram is a right angle,
then the parallelogram is a rectangle.
W
X
Z
Y
Proof: Show that all 4 angles are right angles, then the parallelogram
is a rectangle by the Definition of a Rectangle .
RHOMBUS: a quadrilateral with
four
congruent sides.
Every rhombus is a _____;
Both pairs of opposite sides are ≅
reason:__________________________________________.
Theorem 5-13
The diagonals of a rhombus are
perpendicular .
A
D
X
C
B
Theorem 5-14
Each diagonal of a rhombus
bisects
two angles of the rhombus.
A
D
X
B
C
Proof: Show ∆AXD ≅ ∆AXB ≅ ∆CXD ≅ ∆CXB,
by the SSS Postulate , then use CPCTC to prove …
∠DAX ≅ ∠BAX , ∠DCX ≅ ∠BCX ,
∠ABX ≅ ∠CBX, and ∠CDX ≅ ∠ADX.
Theorem 5-17
If two
consecutive
sides of a parallelogram are congruent,
then the parallelogram is a rhombus.
A
D
X
B
C
Proof: Show that all 4 sides are congruent , then the parallelogram
is a rhombus by the Definition of a Rhombus .
SQUARE: a quadrilateral with four right angles
and four congruent sides.
I’m a
rectangle
too!
I’m a
rhombus
too!
Every square is a _______;
Both pairs of opposite ∠’s are ≅
reason:____________________________________,
or both pairs of opposite sides are ≅ .
__________________________________________
Summary of Special Parallelograms
RECTANGLE:
RHOMBUS:
has four right angles.
has all the properties of a parallelogram.
has diagonals that are congruent.
has four congruent sides.
has all the properties of a parallelogram.
has diagonals that are perpendicular.
has diagonals bisect its angles.
SQUARE: has four right angles and four congruent sides.
has all the properties of a parallelogram.
has diagonals that are both congruent and perpendicular.
has diagonals bisect its angles.
has all the properties of a rectangle and rhombus.
Theorem 5-15
The midpoint of the
hypotenuse
of a right triangle is
equidistant from the three vertices.
A
Given: Right ∆ ABC
X is the midpoint of AB
X
●
Prove: XA = XB = XC
C
circumcenter which is the
The name given to point X is the ____________________,
Perpendicular - Bisector of each side.
intersection of the ________________________________
B
Example #1: State which kind of special quadrilateral this diagram represents.
This is a rhombus, because …
all 4 sides are congruent … definition of a rhombus.
Example #2: State which kind of special quadrilateral this diagram represents.
➤
➤
This is a rectangle, because …
one pair of opposite sides is both parallel and congruent,
so the quadrilateral is a parallelogram.
And a parallelogram with one right angle is a rectangle.
Example #3: State which kind of special quadrilateral this diagram represents.
This is a rectangle, because …
all 4 angles are right angles … definition of a rectangle.
Example #4: State which kind of special quadrilateral this diagram represents.
This is a rhombus, because …
the diagonals bisect each so the quadrilateral is a
parallelogram. And a parallelogram with consecutive
sides congruent is a rhombus.
Assignment
Written Exercises on pages 187 & 188
RECOMMENDED: 1 to 10 (copy & complete the chart)
29, 36 (a), 37
REQUIRED: 13 to 27 odd numbers, 38
You will need graph paper for this assignment.
How can the properties of quadrilaterals
be used to solve real life problems?