chapter 6. formed by 3 or more segments (sides) each side intersects only 2 other sides (one at each...
TRANSCRIPT
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Chapter 6
Quadrilaterals
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Polygons
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What is a Polygon?Formed by 3 or more segments (sides)
Each side intersects only 2 other sides (one at each endpoint)
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What is a Polygon?
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Number of Sides
Name of Polygon
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
n n-gon
Polygons
are
named by
the
number
of sides
they have
What’s in a name
?
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CONVEX
Classifying PolygonsCONCAVE
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Concave or Convex?
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Regular Polygons:Equilateral & Equiangular
Classifying Polygons
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Regular or Irregular?
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Segment that joins 2 non-consecutive vertices.
Diagonals of Polygons
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Diagonals
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Interior Angles of a Quadrilateral Theorem
The Sum of the Measures of the Interior Angles of a Quadrilateral is 360°
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Interior Angles of Quadrilaterals
Solve for x…
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Parallelograms
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QuadrilateralBoth pairs of opposite sides are
parallel
What is a Parallelogram?
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OPPOSITE SIDES are congruent
If a Quadrilateral is a Parallelogram, Then….
OPPOSITE ANGLES are congruent
Theorems about Parallelograms
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CONSECUTIVE ANGLES are supplementary
If a Quadrilateral is a Parallelogram, Then….
DIAGONALS bisect each other
Theorems about Parallelograms
A + B = 180°
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Proving Quadrilaterals are Parallelograms
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If both pairs of opposite sides of a quad. are …
If both pairs of opposite angles of a quad. are …
If an angle of a quad. is supplementary to both of its consecutive angles …
If the diagonals of a quad. bisect each other…
Then, the Quadrilateral
is a Parallelogram.
Prove it!Proving Quadrilaterals are Parallelograms…
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If one pair of opposite sides of a quadrilateral are congruent AND parallel
Then, the Quadrilateral
is a Parallelogram.
Prove it!Proving Quadrilaterals are Parallelograms…
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Describe how to prove that ABCD is a parallelogram given that ∆PBQ ∆RDS and ∆PAS ∆RCQ.
Prove it!Let’s practice….
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Let’s practice….Prove that EFGH is a
parallelogram by showing that a pair of opposite sides are both congruent and parallel.
Use E(1, 2), F(7, 9), G(9, 8), and H(3, 1).
Prove it!Prove that JKLM is a
parallelogram by showing that the diagonals bisect each other.
Use J(-4, 4), K(-1, 5), L(1, -1), and M(-2, -2).
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Quiz 1Sections 1, 2, & 3
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Special Parallelograms
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RhombusA parallelogram with 4 congruent sides
Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides.
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Theorem 6.11:A parallelogram is a rhombus if
and only if its diagonals are perpendicular.
ABCD is a rhombus if and only if AC BD
Rhombus
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Theorem 6.12:A parallelogram is a rhombus if
and only if its diagonals bisect a pair of opposite angles.
ABCD is a rhombus if and only if AD bisects CAB and BDC and BC bisects DCA and ABD
Rhombus
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RectangleA parallelogram with 4 right angles
Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles.
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Theorem 6.13:A parallelogram is a rectangle if
and only if its diagonals are congruent.
ABCD is a rectangle if and only if AC BD
Rectangle
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SquareA parallelogram with 4 congruent sides AND 4 right angles
Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle.
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Special Parallelograms
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Trapezoids
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TrapezoidsQuadrilateral with
only one pair of parallel sides.Parallel sides are
the “bases”Non-parallel sides
are the “legs”Has 2 pairs of base
angles
Base Angles
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Isosceles TrapezoidsShow that RSTV is a
trapezoid…
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Isosceles TrapezoidsLegs are congruent
If mA = 45°, What is the measure of B?
What is the measure of C?
What is the measure of D?
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Isosceles TrapezoidsTheorem 6.14:
If a trapezoid is isosceles, then each pair of base angles is congruent
A D, B C
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Isosceles TrapezoidsTheorem 6.15: (Converse to theorem 6.14)
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid
ABCD is an isosceles trapezoid
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Isosceles TrapezoidsTheorem 6.16:
A trapezoid is isosceles if and only if its diagonals are congruent
ABCD is isosceles if and only if AC BD
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Midsegment Theorem for Trapezoids
(Theorem 6.17)
EF AB, EF DC, EF = ½(AB + DC)
The midsegment of a trapezoid is …Parallel to each
base½ the sum of the
length of the bases
Trapezoids
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Kites
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KitesA quadrilateral that
has two pairs of consecutive congruent sides. Opposite sides are
NOT congruent.
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Theorem 6.18: If a quadrilateral is a kite, then its diagonals are perpendicular
Theorems about
Kites
KT EI
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If KS = ST = 5, ES = 4, and KI = 9, What is the measure of EK?What is the measure of SI?
Practicing Theorems about
Kites
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Theorem 6.19: If a quadrilateral is a kite, then only one pair of opposite angles are congruent
Theorems about
Kites
K M, J L
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If mJ = 70 and mL = 50, What is mM & mK?
Practicing Theorems about
Kites
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Quiz 2Sections 4 & 5
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Special Quadrilaterals
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When you join the midpoints of the sides of ANY quadrilateral, what special quadrilateral is formed? Explain.
On a piece of graph paper… Draw ANY quadrilateralFind and connect the midpoints of each
sideWhat type of Quadrilateral is formed?How do you know?
Special Quadrilaterals
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Let’s prove a quadrilateral is a “special” shape…Use the Definition of the ShapeUse a Theorem
EXAMPE: Show that PQRS is a rhombusHow would you
prove this to be true?
Special Quadrilaterals
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Create a Graphic Organizer showing the relationship between the following
figures…
Isosceles Trapezoid
KiteParallelogramQuadrilaterals
RectangleRhombusSquareTrapezoid
Special Quadrilaterals
Requirements..•Accurate Graphic Organizers
•Each figure should include an picture and description
• Bold, Clear, and Colorful
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Graphic Organizer ExamplesSpecial Quadrilaterals
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Areas
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Area of a RectangleA = bh Find the area of the
rectangle below:
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Area of a ParallelogramA = bh Find the area of the
parallelogram below:
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Area of a TriangleA = ½bh Find the area of the
triangles below:
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Area of a Triangle (again)A = ½bh What is the height
the triangle below:
A = 27 ft2
Base = 9 feet
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Area of a TrapezoidA = ½h(b1 + b2)
Find the area of the trapezoid below:
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Area of a KiteA = ½d1d2
Find the area of the kite below:
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Area of a RhombusA = ½d1d2
Find the area of the rhombus below:
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Quiz 3Sections 6 & 7