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POLYGONS AND QUADRILATERALS

POLYGONS AND QUADRILATERALS. BY: Mariana Beltranena 9-5. Polygon. A polygon is a closed figure with more than 3 straight sides which end points of two lines is the vertex. Parts of Polygons. Sides- each segment that forms a polygon Vertex- common end point of two sides.

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POLYGONS AND QUADRILATERALS

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  1. POLYGONS AND QUADRILATERALS BY: Mariana Beltranena 9-5

  2. Polygon A polygon is a closed figure with more than 3 straight sides which end points of two lines is the vertex.

  3. Parts of Polygons Sides- each segment that forms a polygon Vertex- common end point of two sides. Diagonal- segments that connects any two non consecutive vertices.

  4. 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.

  5. Equilateral and Equiangular Equilateral- all sides are congruent. Equiangular- all angles are congruent.

  6. 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.

  7. 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 convexdoda-gonand find each interior angle. • (12-2)180= 1,800interior measure • 1,800/12= each interior measure

  8. Find each interior measure. (4-2)180=360 c + 3c + c + 3c = 360 8c=360 C= 45 Plug in c m<P = m<R = 45 degrees m<Q= m<S = 135 degrees.

  9. Theorems of Parallelograms and its converse • if a quadrilateral is a polygon then the opposite sides are congruent.

  10. examples

  11. Converse: if both pairs of opposite sides are congruent then the quadrilateral is a polygon EXAMPLES

  12. Theorems of Parallelograms and its converse • If a quadrilateral has one set of opposite congruent and parallel sides then it is a parallelogram.

  13. 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.

  14. 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.

  15. 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.

  16. Theorems of Parallelograms and its converse • If a quadrilateral has consecutive angles which are supplementary, then it is a parallelogram

  17. Converse: If a quadrilateral is supplementary to both of its consecutive angles, then it is a parallelogram.

  18. Theorems of Parallelograms and its converse • If a quadrilateral is a polygon then the opposite angles are congruent.

  19. Converse: If both pair of opposite angles are congruent then the quadrilateral is a parallelogram.

  20. How to prove a Quadrilateral is a parallelogram. • opposite sides are always congruent • opposite angles are congruent • consecutive angles are supplementary • diagonals bisect each other • has to be a quadrilateral and sides parallel • one set of congruent and parallel sides

  21. Square • Is a parallelogram that is both a rectangle and a square. • Properties of a square: • 4 congruent sides and angles • Diagonals are congruent.

  22. rhombus • Is a parallelogram with 4 congruent sides • Properties • Diagonals are perpendicular

  23. 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.

  24. Comparing and contrasting

  25. 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.

  26. 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.

  27. 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.

  28. Kites A kite has two pairs of adjacent angles Diagonals are perpendicular One of the diagonals bisect each other.

  29. 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.

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