Poly gons and Quadrilaterals

1. Polygons and Quadrilaterals Chapter 6 Journal Christian Aycinena 9-5

2. Polygons • A polygon is a • closed figure • formed by three or more segments • each segment meet with another segment at a vertex • no two segments with a common endpoint are collinear.

3. Parts of a Polygon Each segment of a polygon is a side of the polygon. The vertex of the polygon is a common endpoint of two sides. A diagonal is a segment that connects any two nonconsecutive vertices. A C B Side D Vertex F Diagonal E Polygon ABCDEF is a hexagon. Polygon names by the number of its sides.

4. Examples Not a polygon Polygon, Triangle Polygon, octagon Not a polygon Polygon, decagon Polygon, Quadrilateral Polygon, Heptagon Not a polygon Polygon, 17-gon

5. Regular Polygon In an equilateral polygon all the sides are congruent. In a equiangular polygon all the angles are congruent A regular polygon is one that is both equilateral and equiangular, otherwise it is irregular.

6. Concave and Convex A convex polygon has all of the vertices pointing outward. (no internal angles can be more than 180°). A regular polygon is always convex. When one of the vertices is pointing inward the it is a concave polygon. (If there are any internal angles greater than 180° then it is also concave). *Helpful(Think: concave has a "cave" in it)

7. Examples Tell if it is convex or concave. Convex Concave Convex Concave Concave Convex

8. Interior Angles Theorem To find the sum of the interior angles of a convex polygon, draw all the diagonals from one vertex of that polygon. This creates triangles and the sum of the angle measure of all the triangles created equals the sum of the angle measure of the polygon. 1 2 Triangle 1 2 3 Quadrilateral Pentagon 2 3 4 1 2 3 4 And more--- n-gon Hexagon 5 Octagon

9. In each convex polygon, the number of triangles is two less than the number of sides n. So the sum of the angle measure of all these triangles is (n-2)180. Theorem 6-1-1 -The sum of the interior angle measures of a convex polygon with n sides is (n-2)180. So for example in a quadrilateral—(4-2)=2*180=360 is what all the angles measure in a quadrilateral. To find the measure of each angle you divide the sum of the interior angle by the number of sides of the polygon.

10. Examples: 1. Sum of an Pentagon interior angle measures: (n-2)180 (5-2)180 3*180 540° 2. Measure of Nonagon interior angle: (n-2)180 (9-2)180 = 1260° A D 1260°/9 = 140° each interior angle of a nonagon. 3.Find the measure of each interior angle in the fountain. (4-2)180 = 360 B m<A+ m<B + m<C+ m<D = 360 C x + 5x + x +5x =360 12x = 360 x = 30 m<A = m< C = 30° m<B = m<D = 150° x° 5x° 5x° x°

11. Exterior Angle The sum of the exterior angle with one angle at each vertex is always 360° in a convex polygon. (Hint: Usually you add the exterior angle going clockwise in the convex polygon.) 38 36 49 80 40 52 47 41 141 57 139 139 + 141 +80 = 360 36+38+49+52+41+57+47+40 = 360 61 49 54 110 86 54+61+49+110

12. Parallelograms First of all a parallelogram is a quadrilateral with two pairs of parallel sides. It follows some properties that they will be shown on the next page. The symbol of a parallelogram is C B AB || CD, BC|| DA Parallelogram ABCD ABCD A D Here is the first property, by the definition of a parallelogram you can conclude: -If a quadrilateral is a parallelogram, then it has two pairs of parallel sides. Converse: If a quadrilateral has two pairs of parallel sides, then it is as parallelogram.

13. A B ABCD is a parallelogram G F W X After you could state that AB || DC, AD || BC EF || GH, FG || EH WX || ZY, WZ || XY WXYZ is a parallelogram Z Y E EFGH is a parallelogram H D C B A G F After you could state that ABCD is a parallelogram EFGH is a parallelogram WXYZ is a parallelogram W X Z Y E H D C

14. Theorems 1. If a quadrilateral is a parallelogram, then its opposite sides are congruent ( --- opp. Sides =) If a quadrilateral has opposite sides congruent, then it is a parallelogram. Examples: A B ABCD is a parallelogram G You could state that AB = DC, AD = BC EF = GH, FG = EH WX = ZY, WZ = XY F W X WXYZ is a parallelogram Z Y E EFGH is a parallelogram H D C

15. B A G F W You could state that ABCD is a parallelogram EFGH is a parallelogram WXYZ is a parallelogram X Z Y E H D C 2. If a quadrilateral is a parallelogram, then its opposite angles are congruent ( --- opp. <´s=) If a quadrilateral has opposite congruent angles, then it is a parallelogram.

16. A B ABCD is a parallelogram G F You could state that <A = <C, <B = <D <E = <G, <F = <H <W = <Y, <X = <Z W X WXYZ is a parallelogram Z Y E EFGH is a parallelogram H D C B A G F You could state that ABCD is a parallelogram EFGH is a parallelogram WXYZ is a parallelogram W X Z Y E H D C

17. If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. ( --- cons. <´s supp.) If a quadrilateral has supplementary consecutive angles , then it is a parallelogram. You could state that m<A + m< B =180 m<B + m<C =180 m<C + m<D = 180 m<D + m<A =180 m<E + m<F= 180 m<F + m<G= 180 m<G + m<H= 180 m<H + m<E = 180 m<W + m<X=180 m<X + m<Y= 180 m<Y + m<Z = 180 m<Z + m<W= 180 A B ABCD is a parallelogram G F W X WXYZ is a parallelogram Z Y E EFGH is a parallelogram H D C

18. If a quadrilateral is a parallelogram, then its diagonals bisect each other. ( --- diags. Bisect each other) If a quadrilateral has diagonals that bisect each other, then it is a parallelogram. A B You could state that AK = KC DK = KB K D C W X L You could state that WL = LY XL = ZL Z Y

19. E F You could state that EÑ = ÑG HÑ = ÑF Ñ H G A B E F Ñ K D G C H You could state that ABCD is a parallelogram You could state that EFGH is a parallelogram W X L Z Y You could state that WXYZ is a parallelogram

20. Proving Quadrilaterals are Parallelograms • Quadrilaterals have two pairs of opposite sides are parallel • Opposite sides are congruent • Opposite angles are congruent • Consecutive angles are supplementary • Diagonals bisect each other • One set of congruent and parallel sides

21. Examples: 2 pairs of parallel sides Opposite sides congruent Opposite angles congruent 150 30 Consecutive angles are supplementary One set of congruent and parallel sides Diagonals Bisect each other

22. Example:

23. Rhombus, Rectangle, Square All of are parallelograms and contain all the properties a parallelogram contains; also they contain their own properties. Rectangle: is a quadrilateral with 4 right angles Rhombus: is a quadrilateral with four congruent sides Square: is a combination of a rectangle and rhombus(4 rt. <´s and 4 = sides)

24. Theorems • Rectangle Properties • If a quadrilateral is a rectangle, then it is a parallelogram. ( rect.---- ) • If a parallelogram is a rectangle, then its diagonals are congruent. • (rect. --- diags. =) A B After you could state that ABCD is a parallelogram EFGH is a parallelogram WXYZ is a parallelogram D C E F H G W X Z Y

25. A B AC = BD D C E F EG = FH H G W X WY = XZ Z Y

26. Rhombus Properties • If a quadrilateral is a rhombus then it is a parallelogram. (rhombus-- ) • If a parallelogram is a rhombus , then its diagonals are perpendicular. (rhombus—diags. Perpendicular) • If a parallelogram, is a rhombus, then each diagonal bisects a pair of opposite angles. Y A B X Z D C F E W After you could state that ABCD is a parallelogram EFGH is a parallelogram WXYZ is a parallelogram G H

27. Y F A B E X Z G D C H W AC is perpendicular to BD EG is perpendicular to HF YW perpendicular to XZ Y A B F 1 2 1 2 3 4 E 1 2 7 8 X 7 8 Z 3 4 5 6 7 8 3 4 D C 5 6 G 5 6 H <1 = <2 <3= <4 <5 = <6 <7 = <8 <1 = <2 <3= <4 <5 = <6 <7 = <8 W <1 = <2 <3= <4 <5 = <6 <7 = <8

28. Square Properties: • Follows all the properties of a parallelogram, rhombus and rectangle.

29. Trapezoid Quadrilateral with exactly one pair of parallel sides. The parallel sides are called the base and the nonparallel are the legs. Base angles are two consecutive angles whose common side is a base. Base Base Leg Leg Base Angles Isosceles Trapezoid: one pair of congruent legs.

30. Isosceles Trapezoids • If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. • If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. • A trapezoid is isosceles if and only if its diagonals are congruent.

31. Kite • Quadrilaterals with two pairs of congruent consecutive sides. • Properties: • If a quadrilateral is a kite, then its diagonals are perpendicular. • If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

32. KITES