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4.5, 4.6 and 4.8 By Tamago and Faris

4.5, 4.6 and 4.8

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4.5, 4.6 and 4.8. By Tamago and Faris. Today’s Special: QUADRILATERALS!. Quadrilateral – a polygon with four sides Parallelogram – a quadrilateral with two pairs of parallel sides Kite – a quadrilateral with exactly two pairs of adjacent congruent sides - PowerPoint PPT Presentation

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Page 1: 4.5, 4.6 and 4.8

4.5, 4.6 and 4.8

By Tamago and Faris

Page 2: 4.5, 4.6 and 4.8

Today’s Special:QUADRILATERALS!

• Quadrilateral – a polygon with four sides o Parallelogram – a quadrilateral with two pairs of

parallel sides Kite – a quadrilateral with exactly two pairs of adjacent

congruent sides Rectangle – a quadrilateral with four right angles Rhombus – a quadrilateral with four congruent sides Square – quadrilateral with four congruent sides and

angles.o Trapezoid – a quadrilateral with one and only one

pair of parallel sides

Page 3: 4.5, 4.6 and 4.8

Parallelogram Theorems

• The opposite sides of a parallelogram are congruent

• The opposite angles of a parallelogram are congruent

Page 4: 4.5, 4.6 and 4.8

Parallelogram Diagonal Theorems

• A diagonal of a parallelogram divides the parallelogram into two congruent triangles

• The diagonals of a parallelogram bisect each other

           A                                               B

C                                              D

ΔABD ≅ ΔDCA

Page 5: 4.5, 4.6 and 4.8

OMG, I donno if this is a parallelogram!!

It's a parallelogram...• If two pairs of opposite sides of a quadrilateral are

congruent, then the quadrilateral is a parallelogram• If one pair of opposite sides of a quadrilateral are parallel

and congruent, then the quadrilateral is a parallelogram• If the diagonals of a quadrilateral bisect each other, then

the quadrilateral is a parallelogram

Page 6: 4.5, 4.6 and 4.8

Rectangle Theorems

• A rectangle is a parallelogram (and so all parallelogram theorems apply

• The diagonals of a rectangle are congruent

Page 7: 4.5, 4.6 and 4.8

Is It a Rectangle? More Theorems

If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle

Housebuilder Theorem If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle  

Page 8: 4.5, 4.6 and 4.8

• A rhombus is a parallelogram (therefore, all parallelogram theorems apply)

• The diagonals of a rhombus are perpendicular

Theorems of the Rhombus

Page 9: 4.5, 4.6 and 4.8

Help me! I cannot identify this shape as a rhombus!

• If one pair of adjacent sides of a parallelogram are congruent, then the quadrilateral is a rhombus

• If the diagonals of a parallelogram bisect the angles of a parallelogram, then the parallelogram is a rhombus

• If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus

Page 10: 4.5, 4.6 and 4.8

We need squares… NOW!!!!

• A square is a rectangle (so all theorems of rectangles apply)

• A square is a rhombus (and all theorems of rhombi apply)

• The diagonals of a square arecongruent and are the perpendicular bisectors of eachother

Page 11: 4.5, 4.6 and 4.8

Random Kite Slide and Bad Grammar

• The diagonals of a kite are perpendicular

BETWEENNESS POSTULATEGiven points P, Q and R, is PQ + QR = PR, then Q is between P and R on a line

P Q R

Page 12: 4.5, 4.6 and 4.8

Theorems for Triangles! TRIANGLE MIDSEGMENT THEOREM A midsegment of a triangle is parallel to a side of a

triangle and its length is equal to half the length of that side

TRIANGLE INEQUALITY THEOREM The sum of the length of any two sides of a triangle

is greater than the length of the third side

Page 13: 4.5, 4.6 and 4.8

Interesting Facts that May Catch Your Attention so You Don’t Fall Asleep… or Maybe Not.

• The plural of rhombus can either be rhombi OR rhombuses

• Fun little website about rhombi: OMFSM CLICK ME!!

Page 14: 4.5, 4.6 and 4.8

Let’s Prove Some Theorems! A diagonal of a parallelogram divides the parallelogram into two

congruent triangles

Given: parallelogram PLGM with diagonal Prove:

Parallelogram PLGM with diagonal Given

PL GM

∠3 ≅∠2

PM GL

∠1≅∠4

LM LM

ΔLGM ≅ ΔMPL

ΔLGM ≅ ΔMPL

LM

LM

Def. of parallelogram

Alt. int. angles

Def. of parallelogram

Alt. int. angles

Reflexive

ASA

Page 15: 4.5, 4.6 and 4.8

Practice Problems - Can You Do Them? Are these shapes a parallelogram, rectangle,

rhombus or none?

None Parallelogram,

Rhombus

Page 16: 4.5, 4.6 and 4.8

5

60° 40°

7.7

Practice Problems - Can You Do Them? Find the indicated measures. Assume both are

parallelograms

5 80°

7

5

50°

CD

m∠SPR