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## 4.5-4.8 without 4.7

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### 4.5-4.8 without 4.7 Proving quadrilateral properties Conditions for special quadrilaterals Constructing transformations By: Tyler Register and Tre Burse – PowerPoint PPT presentation

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Title: 4.5-4.8 without 4.7

1
4.5-4.8 without 4.7
transformations
• By Tyler Register
• and
• Tre Burse

geometry
2
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

3
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

4
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

5
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
are congruent then the quadrilateral is a rhombus

6
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 mid-segment of a
triangle is parallel to a side of the triangle
and its length is equal to half the length of
than side

7
The Last Theorem Slide
• Betweenness postulate- given the three points P,
Q, and R PQQRPR 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.

8
4-5
Statements Reasons
• Given
• Def of parallelogram
• PLGM is a parallelogram and LM is a diagonal
• Objective- Prove quadrilateral conjectures by
using triangle congruence postulates and
theorems.
• PL II GM

Given parallelogram PLGM with diagonal LM
• Alt. Int. angles
• lt 3 lt 2
• PM II GL
• Def of parallelogram

Prove triangle LGM triangle MPL
• lt1 lt4
• Alt. Int. angles
• LMLM
• reflexive

LGM MPL
• ASA

P
L M G
9
4-6
• There are many theorems in this section that
• The most notable of these theorems is the House
Builder Theorem
• There is also the Triangle Mid- segment Theorem

House Builder Theorem If the diagonals of a
parallelogram are congruent then the
parallelogram is a rectangle
Triangle Mid-segment Theorem A mid-segment 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
10
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 PQQRPR 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.
57gtX X5gt7 X7gt5 2ltXlt13
5
7
X
11
Quiz
• Which of the following are possible lengths of a
triangle?
• A. 14,8,25 B.16,7,23 C.18,8,24
• If one angle of a quadrilateral is a right angle
than the quadrilateral is a ___________.
• Find the measure of the following angles
• ltQ
• ltRPQ
• ltPRQ

Rectangle
60
P
Q S
R
40
80
60
40