The Norm: Why Traditional Schools Are as They Are Chapter 2 By: Ariella Luberto.
M ATH P ROJECT Ariella Lindenfeld & Nikki Colona.
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Transcript of M ATH P ROJECT Ariella Lindenfeld & Nikki Colona.
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MATH PROJECTAriella Lindenfeld & Nikki Colona
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5.1 - PARALLELOGRAMS
A parallelogram is a quadrilateral with both pairs of opposite parallel sides.
Theorem 5-1: Opposite sides of the parallelogram are congruent.
Theorem 5-2: Opposite angles of a parallelogram are congruent.
Theorem 5-3: Diagonals of a parallelogram bisect each other.
G
F
H
E
EFGHEF = HG;FG=EH1
.1
2
3
4EFGH<3 = <4
3.
2. EFGH<E = <G
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EXAMPLE:
X
120
y
7
38
X equals 120 degrees because of OAC which says that Opposite Angles are congruentY equals 22 because it supplementary to 120 So 60 minus 38 is 22
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5.2 – PROVING QUADRILATERALS ARE PARALLELOGRAMS
Theorem 5-4: If both pairs of opposite sides of a quadrilateral are congruent , then the quadrilateral is a parallelogram.
Theorem 5-5: If one pair of opposite sides of a quadrilateral are both parallel, then the quadrilateral is a parallelogram.
Theorem 5-6: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 5-7: If the diagonals of a quadrilateral bisects each other, then the quadrilateral is a parallelogram.
T S
QR
1
23
4
<
<
<<
<<M
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5.3 – THEOREMS INVOLVING PARALLEL LINES
Theorem 5-8: If two lines are parallel, then all points on the line are equidistant from the other line.
L
M
>
>
A B
C D
AB=CD
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5.3 CONTINUED…
Theorem 5-9: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
AX II BY II CZ and AB = BCXY = YZ
A
B
C
X
Y
Z
1 2 3
4 5
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SECTION 3 – THEOREMS INVOLVING PARALLEL LINES
Theorem 5-10
A line that contains the midpoint of one side of a triangle
and is parallel to another side passes through the
midpoint of the third side
Theorem 5-11
The segment that joins the midpoints of two sides of a
triangle
Is parallel to the third side
Is half as long as the third side
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EXAMPLE:
5y+7
6y+3
4x+3
13x+1
2(4x+3) = 13x+18x+6=13x+15=5xX=1
5y+7 = 6y +34=y
X=1, Y=4
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5-4 SPECIAL PARALLELOGRAMS
Rectangle is a quadrilateral with four right angles.
Therefore, every rectangle is a parallelogram.
Rhombus A quadrilateral with four congruent
sides.
Therefore, every rhombus is a parallelogram.
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5-4 SPECIAL PARALLELOGRAMS
Square a quadrilateral with four right angles and four congruent sides. Therefore, every square is a rectangle, a rhombus, and
a parallelogram
*** since rectangles, rhombuses, and squares are parallelograms, they have all the properties of parallelograms.
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5-4 SPECIAL PARALLELOGRAMS
Theorem 5-12
The diagonals of a rectangle are congruent
Theorem 5-13
The diagonals of a rhombus are perpendicular
Theorem 5-14
Each diagonal of a rhombus are perpendicular
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5-4 SPECIAL PARALLELOGRAMS
Theorem 5-15
The midpoint of the hypotenuse of a right triangle is
equidistant from the three vertices
Theorem 5-16
If an angle of a parallelogram is a right angle, then the
parallelogram is a rectangle
Theorem 5-17
If two consecutive sides of a parallelogram are
congruent, then the parallelogram is a rhombus
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5-5 TRAPEZOIDS
Trapezoid a quadrilateral with exactly one pair of
parallel sides
Bases the parallel sides
Legs the other sides
Isosceles Trapezoid a trapezoid with congruent
legs
Base angles are congruent
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5-5 TRAPEZOIDS
Theorem 5-18
Base angles of an isosceles trapezoid are congruent
*** Median (of a trapezoid )the segment that joins the
midpoints of the legs
Theorem 5-19
The Median of a Trapezoid
Is parallel to the bases
Has a length equal to the average of the base lengths
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PRACTICE QUESTIONS:
4x-y
95
7x-4
y1.
75
X z
y
2.
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PRACTICE QUESTION
42
4x+y
26 3x-2y
3.
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PROOF:
Statement Reason
1. Rectangle QRST RKST JQST
1.Given
KS =RTJT = QS
2. OSC
3. RT =QS 3. DB
4. JT = KS 4. substitution
T S
J Q R K