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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
Proving quadrilateral properties Conditions for
special quadrilaterals Constructing
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
  • If one pair of adjacent sides of a quadrilateral
    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
  • 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 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
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