4.5, 4.6 and 4.8
By Tamago and Faris
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
Parallelogram Theorems
• The opposite sides of a parallelogram are congruent
• The opposite angles of a parallelogram are congruent
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
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ΔABD ≅ ΔDCA
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
Rectangle Theorems
• A rectangle is a parallelogram (and so all parallelogram theorems apply
• The diagonals of a rectangle are congruent
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
• A rhombus is a parallelogram (therefore, all parallelogram theorems apply)
• The diagonals of a rhombus are perpendicular
Theorems of the Rhombus
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
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
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
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
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!!
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
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PL GM
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∠3 ≅∠2
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PM GL
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∠1≅∠4
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LM LM
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ΔLGM ≅ ΔMPL
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ΔLGM ≅ ΔMPL
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LM
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LM
Def. of parallelogram
Alt. int. angles
Def. of parallelogram
Alt. int. angles
Reflexive
ASA
Practice Problems - Can You Do Them? Are these shapes a parallelogram, rectangle,
rhombus or none?
None Parallelogram,
Rhombus
5
60° 40°
7.7
Practice Problems - Can You Do Them? Find the indicated measures. Assume both are
parallelograms
5 80°
7
5
50°
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CD
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m∠SPR
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