Download section 4.5-4.8 - Fulton County Schools

4.5-4.8 without 4.7
Proving quadrilateral properties
Conditions for special quadrilaterals
Constructing transformations
By: Tyler Register
and
Tre Burse
geometry
The Vocabulary and Theorems
• A diagonal of a parallelogram divides the
parallelogram into two equal triangles
• Opposite sides of a parallelogram are
congruent
• Opposite angles of a parallelogram are
congruent
• Diagonals of a parallelogram bisect each
other
Theorems cont.
• A rhombus is a parallelogram
• A rectangle is a parallelogram
• The diagonals and sides of a rhombus
form 4 congruent triangles
• The diagonals of a rhombus are
perpendicular
• The diagonals of a rectangle are
congruent
• A square is a rhombus
Theorems cont.
• The diagonals of a square are
perpendicular and are bisectors of the
angles
• If two pairs of opposite sides of a
quadrilateral are congruent then the
quadrilateral is a parallelogram
• If the diagonals of a quadrilateral bisect
each other then the quadrilateral is a
parallelogram
Theorems
• If one angle of a parallelogram is a right
angle then the parallelogram is a rectangle
• If the diagonals of a parallelogram are
congruent then the parallelogram is a
rectangle
• If one pair of adjacent sides of a
quadrilateral are congruent then the
quadrilateral is a rhombus
More Theorems
• If the diagonals of a parallelogram bisect
the angles of the parallelogram then it is a
rhombus
• If the diagonals of a parallelogram are
perpendicular than it is a rhombus
• Triangle mid-segment theorem- A midsegment of a triangle is parallel to a side
of the triangle and its length is equal to
half the length of than side
The Last Theorem Slide
• Betweenness postulate- given the three
points: P, Q, and R PQ+QR=PR then Q is
between P and R on a line.
• The Triangle inequality theorem- The sum
of any two sides of a triangle are greater
than the other side.
4-5
• ObjectiveProve
quadrilateral
conjectures
by using
triangle
congruence
postulates
and
theorems.
M
Given: parallelogram
PLGM with diagonal
LM
Statements
Reasons
•PLGM is a
•Given
parallelogram and
LM is a diagonal
•Def of
• PL II GM parallelogram
•<3=<2
•PM II GL
Prove: triangle LGM=
triangle MPL
•Alt. Int. angles
•Def of
parallelogram
•<1 = <4 •Alt. Int. angles
•LM=LM •reflexive
LGM= MPL •ASA
P
L
G
4-6
Conditions of special quadrilaterals
•There are many
theorems in this
section that state
special cases in
quadrilaterals
•The most notable of
these theorems is
the House Builder
Theorem
•There is also the
Triangle Midsegment Theorem
House Builder Theorem: If the diagonals of
a parallelogram are congruent then the
parallelogram is a rectangle
Triangle Mid-segment Theorem: A midsegment of a triangle is parallel to a side of the
triangle and its length is equal to half the length of
than side
The list of the theorems in 4-6 are on page 5 and 6
4-8
Constructing transformations
• This section has
one theorem and
one postulate
• The Betweenness
postulate
(converse of the
segment addition
postulate) and the
Triangle Inequality
Theorem
The Betweenness postulate: given the three
points: P, Q, and R PQ+QR=PR then Q is between P
and R on a line.
Triangle Inequality Theorem: The sum of any
two sides of a triangle are greater than the other
side.
5
7
X
5+7>X
X+5>7
X+7>5
2<X<13
Quiz
1. Which of the following are possible lengths of a
triangle?
A. 14,8,25
B.16,7,23
C.18,8,24
2. If one angle of a quadrilateral is a right angle than
the quadrilateral is a ___________.
Rectangle
3. Find the measure of the following angles:
<Q= 60
P
Q
<RPQ= 40
<PRQ= 80
60
S
40
R