Quadrilaterals and Polygons

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Quadrilaterals and Polygons
Polygon A plane figure that is formed by three
or more segments (no two of which are collinear),
and each segment (side) intersects at exactly two
other sides one at each endpoint (Vertex).
Which of the following diagrams are polygons?
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Polygons are Named Classified by the Number of
Sides They Have
of Sides Type of Polygon
3
4
5
6
7
of Sides Type of Polygon
8
9
10
12

Octagon Nonagon Decagon Dodagon N-gon
Triangle Quadrilateral Pentagon Hexagon Heptag
on
What type of polygons are the following?
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Convex and Concave Polygons
Convex A polygon is convex if no line that
contains a side of the polygon contains a point
in the interior of the polygon. Concave A
polygon that is not convex
Equilateral, Equiangular, and Regular
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Diagonals and Interior Angles of a Quadrilateral
Diagonal a segment that connects to
non-consecutive vertices.
Theorem 6.1 Interior Angles of a Quadrilateral
Theorem The sum of the measures of the interior
angles of a quadrilateral is 360O mlt1 mlt2
mlt3 mlt4 360o
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Properties of Parallelograms
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Using the Properties of Parallelograms
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Proofs Involving Parallelograms
Plan Show that both angles are congruent to lt2
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Proving Theorem 6.2
Plan Insert a diagonal which will allow us to
divide the parallelogram into two triangles
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Proving Quadrilaterals are Parallelograms
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Concept Summary Proving Quadrilaterals are
Parallelograms

• Show that both pairs of opposite sides are
• Show that both pairs of opposite sides are
• Show that both pairs of opposite angles are
• Show that one angle is supplementary to
• Show that the diagonals
• Show that one pair of opposite sides are both

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Proving Quadrilaterals are Parallelograms
Coordinate Geometry
How can we prove that the Quad is a parallelogram?
1. Slope - Opposite Sides
2. Length (Distance Formula) Opposite sides
same length
3. Combination Show One pair of opposite sides
both and congruent
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Rhombuses, Rectangles, and Squares
Square a parallelogram with four congruent
sides and four right angles
Rhombus a parallelogram with four congruent
sides
Rectangle a parallelogram with four right angles
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Using Properties of Special Triangles
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Using Diagonals of Special Parallelograms
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• Decide if the statement is sometimes, always, or
  never true.
• A rhombus is equilateral.
• 2. The diagonals of a rectangle are __.
• 3. The opposite angles of a rhombus are
  supplementary.
• 4. A square is a rectangle.
• 5. The diagonals of a rectangle bisect each
  other.
• 6. The consecutive angles of a square are
  supplementary.

Always Sometimes

Sometimes Always
Always
Always
32o 86o 66o 35o

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Coordinate Proofs Using the Properties of
Rhombuses, Rectangles and Squares
Using the distance formula and slope, how can we
determine the specific shape of a
parallelogram? Rhombus Rectangle
Square -
Based on the following Coordinate values,
determine if each parallelogram is a rhombus, a
rectangle, or square. P (-2, 3) P(-4,
0) Q(-2, -4) Q(3, 7) R(2, -4) R(6, 4) S(2,
3) S(-1, -3)
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Plan First prove that Triangle PRQ is congruent
to Triangle PRT and Triangle TPQ is congruent
to Triangle TRQ
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Trapezoids and Kites
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Theorems of Trapezoids
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Kites and Theorems about Kites
A kite is a quadrilateral that has two pairs of
consecutive congruent sides, But opposite sides
are NOT congruent. Theorem 6.18 If a Quad is a
Kite, then its diagonals are perpendicular. T
heorem 6.19 If a Quad is a kite then exactly one
pair of opposite angles are congruent
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Using the Properties of a Kite
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Summarizing the Properties of Quadrilaterals
Kites
Parallelograms Trapezoids
Rhombus Squares
Rectangles Isosceles Trap.
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Properties of Quadrilaterals
X X X
X

X
X
X X
X
X X X
X X X X
X X
X
X X X
X X

X
X
X
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Using Area Formulas
Area of a Square Postulate The area of a square
is the square of the length of its side. Area
Congruence Postulate If two polygons are
congruent then they have the same area. Area
Addition Postulate The area of a region is the
sum of the area of its non-overlapping sides.
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Name That Proof
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