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1 6.1 What is a Polygon? Regular Polygon- Polygon Formulas: (n = # of sides) One interior angle: 180(n 2) n Sum of the interior angles of a polygon = 180 (n - 2) Sum of the exterior angles of a polygon = 360 Polygon Names Sides N Name One exterior angle: 360 n Ex: R O U Name the vertices: Name the sides: Name the diagonals containing G: Name 2 consecutive s: Name 2 nonconsecutive sides: G ngles of Polygons 1

2 1. Find the sum of the measures of the angles of a convex polygon with 14 sides. 2. For the given regular polygon, find the measure of each of its interior angles: a) dodecagon b) 16 gon 3. Find the degree measure of each exterior angle of a regular polygon with 20 sides. 4. For the following measures of an angle of a regular polygon, find the number of sides. a) 160 b) The sum of the interior angles of a convex polygon is Find the number of sides. 6. Find the number of sides of a regular polygon if the measure of one of its interior angles Is three times the measure of its adjacent exterior angle. Show all work. Find the sum of the measures of the angles of a convex polygon with the given # of sides For each of the following, the measure of one angle of a regular convex polygon is given. Find the # of sides For each of the following, the number of sides of a regular polygon is given. Find the measure of each angle Find the degree measure of one exterior angle for a regular polygon with the given # of sides The sum of the measure of the interior angles of a convex polygon is lassify the polygon. 16. The measure of one exterior angle of a regular polygon is 45. lassify the polygon. 17. Find the number of sides of a regular polygon, if the measure of one of its interior angles equals the measure of its adjacent exterior angle. 18. Find the number of sides of a regular polygon, if the measure of one of its interior angles equals twice the measure of the adjacent exterior angle. 19. lassify the regular polygon, if the measure of one of its interior angles equals one-half the measure of the adjacent exterior angle. 20. If the sum of the measures of six interior angles of a heptagon is 755, what is the measure of the remaining angle? Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides For a polygon to be regular, it must be both equiangular and equilateral. Name the only type of polygon that must be regular if it is equiangular. Tell whether each polygon is regular or irregular. Then tell whether it is concave or convex Find the sum of the interior angle measures of a 14-gon. 9. Find the measure of each interior angle of hexagon EF. 10. Find the value of n in pentagon PQRST. 2

3 Quadrilateral is any 4-sided polygon. The sum of interior angles for every quadrilateral is 360. Kites 2 pairs of congruent consecutive sides (unequal) Te diagonals are perpendicular Rectangles ll properties of parallelogram Four right angles iagonals are congruent & bisect each other onsecutive sides are perpendicular Squares ll properties of the parallelogram, the rectangle and the rhombus Parallelograms Opposite sides are congruent Opposite sides are parallel Opposite angles are congruent onsecutive angles are supplementary The diagonals bisect each other Trapezoids Has only 1 pair of parallel sides (base 1 & base 2) Non-parallel sides are called legs Rhombus ll the properties of parallelogram Four congruent sides onsecutive sides are congruent iagonals bisect opposite angles iagonals perpendicular & bisect each other Example E 12 m< = 27 m< = 38 m< = 63 m< = 52 7 Find each measure. True/False 1. Every parallelogram is a quadrilateral. 2. Every quadrilateral is a parallelogram. 3. ll angles of a parallelogram are congruent. 4. Opposite sides of a parallelogram are always congruent. 5. In PEX, P // PX. 6. In RY, R Y. 7. In TO, T and O bisect each other. 3

4 The following figures are s. 1. Given: O = 2x + 5y S TO = 18 ON = 30 O SO = 2x + 2y Find x and y. T N P S Example 2. Given: mp = 85 mps = 130 mls = (x y) msl = (x + y) oordinate Geometry: Show that (2, -1) (1, 3) (6, 5) and (7, 1) are vertices of a parallelogram. Method 1: Find the slopes (opposite sides should be parallel) L Method 2: Find lengths (opposite sides should be congruent) Method 3: One pair of opposite sides are parallel and congruent Proving a Quadrilateral is a parallelogram quadrilateral is a parallelogram if: 1. both pairs of opposite sides are parallel (by definition)

5 5

6 Rectangles efinition: rectangle is a quadrilateral with. efinition: rectangle is a parallelogram with. Prove the diagonals are in a Rectangle: Given: Rectangle Prove: To Prove that a quadrilateral is a rectangle, prove that: 1) It is a quadrilateral with. 2) It is a parallelogram with. 3) It is a parallelogram with. Find all interior angle measures given rectangle STN. 31 S O Which of the following quadrilaterals are rectangles? Justify your answer N T For 4 10, is a parallelogram. From the information given, tell whether is a rectangle. 4. Given: 5. Given: 6. Given: is a right angle. 7. Given: 8. Given: 9. Given: 10. Given: ; is a right angle 11. Find x and y Given: iagonals RP and SQ of rectangle PQRS meet at M. If PM = x + 3y, SM = 4y 2x and RM = 20. 6

7 Rhombus efinition: quadrilateral is a rhombus iff. efinition: parallelogram is a rhombus iff. H R O Mark the rhombus. How many s? What must be true about HO? Therefore, diagonals must be. Theorem: parallelogram is a rhombus iff. M To Prove that a quadrilateral is a rhombus, prove that: 1) It is a quadrilateral with. 2) It is a parallelogram with. 3) It is a parallelogram with. 4) It is a parallelogram with. 5) Find all interior angles of the following rhombus. 23 6) Given: ZY; ZY X ; 1 2 Prove: ZY is a rhombus X Z Y Which of the following are rhombuses? Justify each answer For 4 10, is a parallelogram. From the info. Given tell whether is a rhombus. 4. Given: 5. Given: 6. Given: is a right angle 7. Given: 8. Given: ; is a right angle 9. Given: 10. Given: 11. In rhombus, m = 3x 5 and m = 11x 3. Find the measures of all the angles of the rhombus. 12. In parallelogram, = 17x 3, = 13x + 5, and = 4x Find the lengths of the sides of parallelogram. What type of parallelogram is? 7

8 PROOFS: 1. Given: Rhombus ; FE // Prove: EF E F 2. Given: Rhombus Prove:

9 parallelogram is a square iff it has one right angle and 2 adjacent sides. square is both a and a. square has all of the properties of a,, and. To prove a quadrilateral is a square, prove that: 1) It is a rectangle with. 2) It is a rectangle with. 3) It is a rectangle with. 4) It is a rhombus with. 5) It is a rhombus with. 6) It is a parallelogram with. omplete the following. 1. Every rectangle is also a. 2. Every rhombus is also a. 3. Every square is also a, and a. 4. with diagonals is a or a. 5. with diagonals is a or a. 6. whose diagonals are the bisectors of each other is a or a. True or False. 7. ll rhombi are parallelograms. 8. Some rectangles are squares. 9. ll parallelograms are rectangles. 10. Some rhombi are rectangles. 11. ll rectangles are squares. 12. ll squares are rectangles. 13. Some squares are rectangles. 14. Given: Square HIJK and equilateral ILJ. Find mhli. H I L 15. In Square, diagonals and meet at E. Find m. K J 16. The rectangle to the right is divided in squares. The 2 largest are 69 and 72 units on a side. How long is each side of each square? Use square and the given information to find each value. 14. If me = 3x, find x. 15. If m = 9x, find x. 16. If = 2x + 1 and = 3x 5, find 17. If m = y and m = 3x, find x and y. 18. If = x 2-15 and = 2x, find x E 9

10 6.6 Kites and Trapezoids y 6x 3x 2 Find the value(s) of the variable(s) in each kite (3x+5) 1 6 (x+6) 15y (2x-4) y (4x- 30) (2y-20) x an two angles of a kite be as follows? 7. opposite and acute 8. consecutive and obtuse 9. opposite and supplementary 10. consecutive and supplementary 11. opposite and complementary 12. consecutive and complementary 13. The perimeter of a kite is 66 cm. The length of one of its sides is 3 cm less that twice the length of another. Find the length of each side of the kite. 14. etermine whether the points (4, 5), (-3, 3), (-6, -13) and (6, -2) are the vertices of a kite. Show your work. Trapezoids efinition: trapezoid is a quadrilateral with exactly two parallel sides. Parts of a trapezoid: Leg ase Leg ase Leg 10 ase Leg ase

11 Isosceles Trapezoid: trapezoid with congruent legs. Every trapezoid contains two pairs of consecutive angles that are supplementary. Example 1: Given the trapezoid HLJK H L If the m J 65 and the K J m K 95, the measure of angles H and L. Theorem: The base angles of an isosceles trapezoid are congruent. Theorem: The diagonals of an isosceles trapezoid are congruent. Example 2: Use Isosceles Trapezoid with length of =. // a. m = 75. Find the m. b. = 40. Find. c. If m 6x 25 and m 8x 15, find the measures of angle and. efinition: m altitude is a line segment from one vertex of one base of the trapezoid and perpendicular to the opposite base. Theorem: The length of the median of a trapezoid equals one-half the sum of the bases. m Example 3: Find the missing measures of the given trapezoid. 1 2 b 1 b 2 7 I a. mir b. YR c. R d X Y R

12 Example 4: HJKL is an isosceles trapezoid with bases HJ and LK, and median RS. Use the given information to solve each problem. a. LK = 30 find LK L K HJ = 42 find RS c. RS = x + 5 R S b. RS = 17 HJ = 14 HJ + LK = 4x + 6 find RS H J Example 5: 5x + 12 Find the length of each side of the isosceles trapezoid below. 6x x 14x 12

13 TRPEZOI PRTIE I. Fill in the blanks: 1) trapezoid is a quadrilateral with exactly one pair of _?_ opposite sides. 2) If the non-parallel sides of a trapezoid are, then it is called a(n) _?_ trapezoid. 3) The parallel sides of a trapezoid are called its _? 4) nd the non-parallel sides are called the _? 5) The segment that joins the midpoints of the non-parallel sides of a trapezoid is called the _? 6) and is parallel to the _?_ of the trapezoid. II. nswer the following. Show your work to get credit. 7) 8) 9) x x y 9x z 3x-5 x = x = y = x = z = m = m = 10 x 10) 11) 12) x x w y z w = x = x = x = y = 13

14 LWYS, SOMETIMES, OR NEVER Part The diagonals of a quadrilateral bisect each other. 2. If the measures of 2 angles of a quadrilateral are equal, then the quadrilateral is a parallelogram. 3. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram. 4. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. 5. To prove a quadrilateral is a parallelogram, it is enough to show that one pair of opposite sides is parallel. 6. The diagonals of a rectangle are congruent. 7. The diagonals of a parallelogram bisect the angles. Part 2. always-sometimes-never 1. square is a rhombus. 2. The diagonals of a parallelogram bisect the angles of the parallelogram. 3. quadrilateral with one pair of sides congruent and one pair parallel is a parallelogram. 4. The diagonals of a rhombus are congruent. 5. rectangle has consecutive sides congruent. 6. rectangle has perpendicular diagonals. 7. The diagonals of a rhombus bisect each other. 8. The diagonals of a parallelogram are perpendicular bisectors of each other. 14

15 15

16 Square wksht Property 1. It has Four Sides 2. Opposite Sides are Parallel 3. Opposite Sides are ongruent 4. Opposite ngles are ongruent 5. iagonals isect Each Other 6. iagonal Forms Two ongruent Triangles 7. iagonals are ongruent 8. iagonals are Perpendicular to each other 9. iagonal isects Two Opposite ngles 10. ll ngles are Right ngles 11. ll Sides are ongruent 12. Two Sides are ongruent and djacent 13. onsecutive ngles re Supplementary 14. onsecutive angles are congruent. 15. It is equilateral. 16. It is equiangular. 17. onsecutive (adjacent) sides are perpendicular. 18.iagonals are congruent, perpendicular, and bisect each other. lgebraic Formulas Used to etermine the Type of Quadrilateral Quadrilateral Parallelogram Rectangle Rhombus Square istance Formula-**use to determine if opposite sides are 2 2 d x x y y Midpoint Formula **use to determine if diagonals bisect each other(they have to have the some midpoint) x x y y, Slope Formula-**use to determine if opposite sides are parallel m y x 2 2 y 1 x 1 Honors Geometry oordinate onnection Notes 16

17 To Show that a quadrilateral is a Parallelogram Method 1: oth pairs of opposite sides are congruent (find distance) Method 2: oth pairs of opposite sides are parallel (find slope) Method 3: One pair of opposite sides are both parallel and congruent (find distance and slope) To show that a quadrilateral is a Rhombus ****FIRST show that it is a parallelogram**** Method 1: ll 4 sides are congruent Method 2: iagonals are perpendicular (find slope of diagonals) To show that a quadrilateral is a Rectangle ****FIRST show that it is a parallelogram**** Method 1: ll angles are right angles Method 2: iagonals are congruent (find distance of diagonals) To show that a quadrilateral is an Isosceles Trapezoid Graph first o Legs are congruent (find distance) o ases are parallel (find slope) iagonals are congruent To show that a quadrilateral is a Kite Graph first o Two pairs of consecutive congruent sides that are not congruent to each other (find the distance) Honors Geometry oordinate onnection 1. Given: (-1,-6), (1,-3), (11,1) and (9,-2) Show that Quad. is a parallelogram. 2. Given: E(-4,1), F(2,3), G(4,9) and H(-2,7) a) Show that EFGH is a rhombus. b) Use slopes to verify that the diagonals are perpendicular. 3. Given: R(-4,5), S(-1,9), T(7,3) and U(4,-1) a) Show that RSTU is a rectangle. b) Use the distance formula to verify that the diagonals are. 4. Given: N(-1,-5), O(0,0), P(3,2) and Q(8,1) a) Show that NOPQ is an isosceles trapezoid. b) Show that the diagonals are. More examples: 1. Given: with (2,6), (3,-5), and (-1,7). If M is the midpoint of and N is the midpoint of, show MN //. 2. Given: Trapezoid with (4,0), (8,0), (0,2) and (0,4). Find the length of the median. 3. Given: Rectangle with (6,-4), (6,2), (3,2) and (3,-4). Write the equations of the lines that contain the diagonals in standard form. 4. Given: Quad. with (3.2), (8,1), (7,6) and (2,7). What name best describes the quadrilateral? Prove (explain) your answer. 5. RHOM is a rhombus. R(1,3), M(14,3), H(6,15) a) Find the coordinate of O. b) Find the slopes of HM and RO. c) What does part b verify? H O 17 R M

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