6.1 What is a Polygon? A closed plane figure formed by segments that intersect only at their endpoints Regular Polygon- a polygon that is both equiangular and equilateral Unit 6 Polygons and Quadrilaterals Side one of the line segments that determine the polygon. Vertex- The intersection of two sides. The plural is vertices. Interior angle Diagonal Side vertex Diagonal- a segment connecting two nonconsecutive vertices. Interior Angle- An angle formed by two adjacent sides Exterior angle- formed by one side of a polygon and the extension of an adjacent side Polygon Names: # sides Name 3 Triangle 4 Quadrilateral Draw hexagon RAPTOE 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Undecagon 12 Dodecagon 13 Tridecagon 14 Tetradecadon 15 Pentadecagon Name the vertices: Name the sides: Name the diagonals containing G Name 2 consecutive s Name 2 consecutive sides n n- gon 1
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1. 2. 3. 4. 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 5 6. 7. regular or irregular. Then tell whether it is concave or convex. 8. Find the sum of the interior angle measures of a convex heptagon. 9. Find the measure of each interior angle of hexagon ABCDEF. 10. Find the value of n in pentagon PQRST. Polygon Formulas:(n= # of sides) Sum of the interior angles of a convex polygon = (n-2) 180 Sum of the exterior angles of a convex polygon = 360 One interior angle of a convex polygon = 180( n 2) n One exterior angle of a convex polygon = 360 n Polygon Practice: 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) 140 2
5. The sum of the interior angles of a convex polygon is 2520. 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. Find the sum of the measures of the angles of a convex polygon with the given # of sides. 1. 17 2. 20 3. 12 For each of the following, the measure of one angle of a regular convex polygon is given. Find the # of sides. 4. 150 5. 120 6. 156 For each of the following, the number of sides of a regular polygon is given. Find the measure of each angle. 7. 4 8. 8 9. 10 Find the degree measure of one exterior angle for a regular polygon with the given # of sides 10. 8 11. 5 12. 13 13. 10 14. 6 15. The sum of the measure of the interior angles of a convex polygon is 1260. Classify the polygon. 16. The measure of one exterior angle of a regular polygon is 45. Classify 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. Classify 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? 6.2-6.3 A Quadrilateral is any 4-sided polygon. The sum of interior angles for every quadrilateral is 360º Example 1. m DCA = 27 Solve for x and y. 6 B 7 4 E 12 C m CAD = 38 m ABD = 63 Find each measure. A D 3
True/False 1. Every parallelogram is a quadrilateral. 2. Every quadrilateral is a parallelogram. 3. All angles of a parallelogram are congruent. 4. Opposite sides of a parallelogram are always congruent. 5. In APEX 6. In CARY, AP // PX. CR AY 7. In TOAD, TA and OD bisect each other 4
Proving a Quadrilateral is a parallelogram: A quadrilateral is a parallelogram if: (one of these conditions is necessary) 1. Both pairs of opposite sides are parallel (by definition) 2. One pair of opposite sides are // and 3. Both pairs of opposite sides are 4. Both pairs of opposite angles are 5. One is supplementary to both of its consecutive s. 6. The diagonals bisect each other State whether the given information is sufficient to support the statement, Quadrilateral ABCD is a parallelogram. If the information is sufficient, state the reason. 5
Proofs! 1) Given: 1 2; 3 4 Prove: QRST is a parallelogram Q 1 4 R S 2) Given: 1 2, AE EC Prove: ABCD is a Parallelogram 3) Given: PQRS PJ RK R Prove: SJ QK 4) Given: QRST; QXYZ Prove: Y S Z Q Y X R T S 5) Given: AE CD, DBC C, A DBC Prove: ABDE is a parallelogram 6) 6
6.4-6.5 Rectangles Definition: A rectangle is a quadrilateral with 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. Which of the following quadrilaterals are rectangles? Justify your answer. 1. 2. 3. For 4 10, ABCD is a parallelogram. From the information given, tell whether ABCD is a rectangle. 4. Given: AD AB 5. Given AC DB A B 6, Given: BCD is a right angle. 7. Given: AC BD 8. Given: AC BD; ADC is a right angle 9. Given: ADC BCD 10.Given: DAC BAC D C 11.Find x and y Given: Diagonals RP and SQ of rectangle PQRS meet at M, and PM = x + 3y, SM = 4y 2x and RM = 20. 7
Rhombus Definition: A quadrilateral is a rhombus iff. To prove that a parallelogram is a rhombus: A parallelogram is a rhombus iff. A parallelogram is a rhombus iff. A parallelogram is a rhombus iff. H The m RHB = 23º R B O Find the measure of the remaining interior angles. M Mark the rhombus. How many s? What must be true about HBO? Which of the following are rhombuses? Justify each answer. 1. 2. 3. For 4 10, ABCD is a parallelogram. From the info. Given tell whether ABCD is a rhombus. 4. Given: AB AD B 5. Given: AC DB 6. Given: BCD is a right angle A C 7. Given: AC BD 8. Given: AC BD; ADC is a right angle D 9. Given: ADC BCD 10. Given: DAC BAC 8
11. In rhombus ABCD, m ABD = 3x 5 and m BAC = 11x 3. Find the measures of all the angles of the rhombus. 12. In parallelogram ABCD, AB = 17x 3, BC = 13x + 5, and CD = 4x + 23. Find the lengths of the sides of parallelogram ABCD. What type of parallelogram is ABCD? A parallelogram is a square iff it has one right angle and 2 adjacent sides. A square is both a and a. 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. A B 3) It is a rectangle with. E 4) It is a rhombus with. 5) It is a rhombus with. D 6) It is a parallelogram with. True or False. 7. All rhombi are parallelograms. 8. Some rectangles are squares. 9. All parallelograms are rectangles. 10. Some rhombi are rectangles. 11. All rectangles are squares. 12. All squares are rectangles. 13. Some squares are rectangles. C Use square ABCD and the given information to find each value. A B 14. If m AEB = 3x, find x. 15. If m BAC = 9x, find x. 16. If AB = 2x + 1 and CD = 3x 5, find BC 17. If m DAC = y and m BAC = 3x, find x and y. 18. If AB = x 2-15 and BC = 2x, find x. D E C 9
Quadrilaterals SOME parallelograms trapezoids kites other quadrilaterals rectangles rhombi isosceles trapezoids squares ALL 6.6 Kites and Trapezoids A kite is a quadrilateral with two distinct pairs of adjacent sides. Properties: The diagonals are Exactly one pair of opposite angles are The long diagonal bisects the short diagonal The long diagonal bisects the opposite s 1. 2. 3. 3 35º 1 2 54º 100º yº 9xº Find the value(s) of the variable(s) in each kite. 4. 5. 6. (3x+5)º xº (x+6)º 15y 2xº (2x-4)º yº (4x-30)º (2y-20)º 35º 5 1 2 3 4 6 7 Can 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 than twice the length of another. Find the length of each side of the kite. 10
Trapezoids A trapezoid is a quadrilateral with exactly two parallel sides. Isosceles Trapezoid: A trapezoid with congruent legs. Theorem: The base angles of an isosceles trapezoid are congruent. Theorem: The diagonals of an isosceles trapezoid are congruent. Every trapezoid contains two pairs of consecutive angles that are supplementary. Leg Base Base Leg Example 1: Given the trapezoid HLJK. If the m J = 65º and the m K = 95º, find the measure of angles H and L. Example 2: Use Isosceles Trapezoid ABCD with length of AD = BC. // D C a. m DAB = 75º. Find the m ADC. b. AC = 40. Find BD. c. If m A=6x+25 and m B = 8x +15, find the measures of angle C and D. A B Midsegment A line segment connecting the midpoints of the legs of a trapezoid. The midsegment is parallel to the bases. Theorem: The length of the midsegment of a trapezoid equals one-half the sum of the bases. 1 m b1 b2 2 Example 3: HJKL is an isosceles trapezoid with bases HJ and LK, and midsegment RS. Use the given information to solve each problem L K a. LK = 30, HJ = 42, find RS b. RS = 17, HJ = 14, Find LK. c R H RS = x + 5, HJ + LK = 4x + 6 find RS Example4: Find the length of each side of the isosceles trapezoid below. J S 11
Algebraic Formulas Used to Determine the Type of Quadrilateral To Show that a quadrilateral is a Parallelogram a. Method 1: Both pairs of opposite sides are congruent (find distance) b. Method 2: Both pairs of opposite sides are parallel (find slope) c. 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**** d. Method 1: All 4 sides are congruent e. Method 2: Diagonals are perpendicular (find slope of diagonals) To show that a quadrilateral is a Rectangle ****FIRST show that it is a parallelogram**** f. Method 1: All angles are right angles g. Method 2: Diagonals are congruent (find length of diagonals) To show that a quadrilateral is an Isosceles Trapezoid h. Graph first o Legs are congruent (find distance) o Bases are parallel (find slope) i. Diagonals are congruent To show that a quadrilateral is a Kite j. Two pairs of distinct consecutive congruent sides (find the distance) 12
Practice Determining Type of Quadrilateral (Most Precise Name) 1) Given quadrilateral ABCD with coordinates A (-1, -2), B (4, 4), C (10, -1), and D (5, -7). Is quad ABCD a rectangle? (Note: Use the slope formula.) Slope of AB = Slope of BC = Slope of CD = Slope of AD = 2) The coordinates of quadrilateral QRST are: Q (-2, -1), R (-1, 2), S (2, 3), and T (1, 0). a) Find the slopes of the diagonals of quad QRST. Are they perpendicular? Slope of QS = Slope of RT = b) Find the midpoints of each of the diagonals. Midpoint of QS = (, ) Midpoint of RT = (, ) Do they bisect each other? Why or why not? c) What are all the possible classifications for quad QRST? d) The most precise name? 3) Given: A (-1,-6), B (1,-3), C (11, 1) and D (9,-2) Show that Quad. ABCD is a parallelogram. 4) Given: EFGH is a parallelogram with E(-4,1), F(2,3), G(4,9) and H(-2,7) Show that EFGH is a rhombus. 5) Given: RSTU is a parallelogram with R (-4, 5), S (-1, 9), T (7, 3) and U (4,-1) Show that RSTU is a rectangle. 13
6) Given: Quad. ABCD with A (6,-4), B (6, 2), C (3, 2) and D (3,-4). Is this a parallelogram? For #7 9, use slope, midpoint and/or the distance formulas to determine the most precise name for the quadrilateral with the given vertices. 7) A (-4, 3), B (-4, 8), C (3, 10) D (3, 5) 8) A (-3, 7), B (1, 10), C (1, 5), D (-3, 2) 9) A (6, -5), B (3, 10), C (0, -5), D (3, -9) 10) A (-3, -3), B (3, 4), C (5, 0), D (-4, -1) 11) A (6,0), B (0, 6), C (-6, 0), D (0, -6) 14