by: mariana beltranena 9-5 polygons and quadrilaterals
TRANSCRIPT
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BY: MARIANA BELTRANENA 9-5
POLYGONS AND QUADRILATERALS
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Polygon
A polygon is a closed figure with more than 3 straight sides which end points of two lines is the vertex.
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Parts of Polygons
Sides- each segment that forms a polygonVertex- common end point of two sides.Diagonal- segments that connects any two non
consecutive vertices.
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Convex and Concave polygons
Concave- any figure that has one of the vertices caved in.
Convex- any figure that has all of the vertices pointing out.
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Equilateral and Equiangular
Equilateral- all sides are congruent.Equiangular- all angles are congruent.
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Interior Angles Theorem for Polygons
The sum of the interior angles of a polygon equals the number of the sides minus 2, times 180. (n-2)180. For each interior angle it is the same equation divided by the number of sides.
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examples
Find the sum of the interior angle measure of a convex octagon and find each interior angle. (n-2)180 (8-2)180= 1,080. The interior angles
measure 1,080 degrees. 1080/8=135. Each interior angle measures 135 degrees.
Find the sum of the interior angle measure of a convex doda-gon and find each interior angle. (12-2)180= 1,800interior measure 1,800/12= each interior measure
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Find each interior measure.(4-2)180=360c + 3c + c + 3c = 3608c=360C= 45Plug in cm<P = m<R = 45 degreesm<Q= m<S = 135 degrees.
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Theorems of Parallelograms and its converse
if a quadrilateral is a polygon then the opposite sides are congruent.
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examples
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• Converse: if both pairs of opposite sides are congruent then the quadrilateral is a polygon
EXAMPLES
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Theorems of Parallelograms and its converse
If a quadrilateral has one set of opposite congruent and parallel sides then it is a parallelogram.
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This quad. Is a parallelogram because it has one pair of opposite sides congruent and one pair of parallel sides.
This figure is also a parallelogram because it has two pairs of parallel and congruent opposite sides.
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• Converse: If one pair of opposite sides of a quadrilateral are parallel and congruent, then it is a parallelogram
• Examples: Given- KL ll MJ, KL congruent to MJ• Prove- JKLM is a parallelogram• Proof: It is given that KL congruent to MJ. Since KL
ll MJ, <1 congruent to <2 by the alternate interior angles theo. By the reflexive property of congruence, JL congruent JL. So triangle LMJ by SAS. By CPCTC, <3 congruent <4 and JK ll to LM by the converse of the alternate interior angles theo. Since the opposite sides of JKLM are parallel, JKLM is a parallelogram by deff.
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Define if the quadrilateral must be a parallelogram.Yes it must be a parallelogram because if one pair of opposite sides of a quadrilateral are parallel and congruent, then it is a parallelogram.
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Theorems of Parallelograms and its converse
If a quadrilateral has consecutive angles which are supplementary, then it is a parallelogram
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• Converse: If a quadrilateral is supplementary to both of its consecutive angles, then it is a parallelogram.
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Theorems of Parallelograms and its converse
If a quadrilateral is a polygon then the opposite angles are congruent.
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• Converse: If both pair of opposite angles are congruent then the quadrilateral is a parallelogram.
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How to prove a Quadrilateral is a parallelogram.
opposite sides are always congruentopposite angles are congruentconsecutive angles are supplementarydiagonals bisect each otherhas to be a quadrilateral and sides parallelone set of congruent and parallel sides
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Square
Is a parallelogram that is both a rectangle and a square. Properties of a square:
4 congruent sides and angles Diagonals are congruent.
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rhombus
Is a parallelogram with 4 congruent sides Properties
Diagonals are perpendicular
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rectangle
Is any parallelogram with 4 right angles.Theorem: if a a quadrilateral is a rectangle
then it is a parallelogram. Properties
Diagonals are congruent 4 right angles.
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Comparing and contrasting
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Trapezoids
Is a quadrilateral with one pair of sides parallel and the other pair concave.
There are isosceles trapezoids with the two legs congruent to each other.
Diagonals are congruent.Base angles are congruent.
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Trapezoid theorems
Theorems: 6-6-3: if a quadrilateral is an isosceles trapezoid, then
each pair of base angles are congruent. 6-6-4: a trapezoid has a pair of congruent base angles,
then the trapezoid is isosceles. 6-6-5: a trapezoid is isosceles if and only if its
diagonals are congruent.
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Trapezoid midsegment theorem: the midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases.
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Kites
A kite has two pairs of adjacent anglesDiagonals are perpendicularOne of the diagonals bisect each other.
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Kite theorems
6-6-1: if a quad. is a kite, then its diagonals are perpen
6-6-2: if a quad is a kite, then exactly one pair of opposite angles are congruent.