You can use the Polygon Interior Angles Sum Theorem to find the sum of the interior angles of a polygon and to find missing measures in polygons.

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3 Geometry Section 6.1 Notes: Angles of Polygons A diagonal of a polygon is a segment that connects any two vertices. The vertices of polygon PQRST that are not consecutive with vertex P are vertices R and S. Therefore, polygon PQRST has two diagonals from vertex P, PR and PS. Notice that the diagonals from vertex P separate the polygon into three triangles. The of the angle measures of a polygon is the of the angle measures of the triangles formed by drawing all the possible diagonals from one vertex. Since the sum of the angle measures of a triangle is 180, we can make a table and look for a pattern to find the sum of the angle measures for any convex polygon. This leads to the following theorem: You can use the Polygon Interior Angles Sum Theorem to find the sum of the interior angles of a polygon and to find missing measures in polygons. Example 1: a) Find the sum of the measures of the interior angles of a convex nonagon. b) Find the measure of each interior angle of parallelogram RSTU. 73

4 Recall from Lesson 1.6 that in a regular polygon, all of the interior angles are congruent. You can use this fact and the Polygon Interior Angle Sum Theorem to find the interior angle measure of any regular polygon. Example 2: A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the measure of one of the interior angles of the pentagon. Given the interior angle measure of a regular polygon, you can also use the Polygon Interior Angles Sum Theorem to find a polygon s number of sides. Example 3: a) The measure of an interior angle of a regular polygon is 150. Find the number of sides in the polygon. b) The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. Does a relationship exist between the number of sides a convex polygon and the sum of its exterior angle measures? Examine the polygons below in which an exterior angle has been measured at each vertex. Did you notice that the sum of the exterior angle measures in each case is 360? This suggests the following theorem: Example 4: a) Find the value of x in the diagram. b) Find the measure of each exterior angle of a regular decagon. 74

5 Geometry Section 6.1 Worksheet Name: For numbers 1 3, find the sum of the measures of the interior angles of each convex polygon gon gon gon For numbers 4 6, the measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon For numbers 7 & 8, find the measure of each interior angle For numbers 9 14, find the measures of an exterior angle and an interior angle given the number of sides of each regular polygon. Round to the nearest tenth, if necessary Crystals are classified according to seven crystal systems. The basis of the classification is the shapes of the faces of the crystal. Turquoise belongs to the triclinic system. Each of the six faces of turquoise is in the shape of a parallelogram. Find the sum of the measures of the interior angles of one such face. 75

6 16. In the Uffizi gallery in Florence, Italy, there is a room built by Buontalenti called the Tribune (La Tribuna in Italian). This room is shaped like a regular octagon. What angle do consecutive walls of the Tribune make with each other? 17. Jasmine is designing boxes she will use to ship her jewelry. She wants to shape the box like a regular polygon. In order for the boxes to pack tightly, she decides to use a regular polygon that has the property that the measure of its interior angles is half the measure of its exterior angles. What regular polygon should she use? 18. A theater floor plan is shown in the figure. The upper five sides are part of a regular dodecagon. Find m Archeologists unearthed parts of two adjacent walls of an ancient castle. Before it was unearthed, they knew from ancient texts that the castle was shaped like a regular polygon, but nobody knew how many sides it had. Some said 6, others 8, and some even said 100. From the information in the figure, how many sides did the castle really have? 20. In Ms. Rickets math class, students made a polygon path that consists of regular polygons of 3, 4, 5, and 6 sides joined together as shown. a) Find m 2 and m 5. b) Find m 3 and m 4. c) What is m 1? 76

7 Geometry Section 6.2 Notes: Parallelograms A parallelogram is a quadrilateral with both pairs of parallel. To name a parallelogram, use the symbol. In ABCD, BC AD and AB DC by definition. Other properties of parallelograms are given in the theorems below. Example 1: In parallelogram ABCD, suppose m B = 32, CD = 80 inches, and BC = 15 inches. a) Find AD. b) Find m C. c) Find m D. 77

8 The diagonals of a parallelogram have special properties as well. Example 2: a) If WXYZ is a parallelogram, find the value of r. WX because.. WX = because.. 4r = because. r = because. b) Find the value of s. c) Find the value of t. WXY because. YWX because.. m YWX = because.. 2t = because.. t = because.. 78

9 You can use Theorem 6.7 to determine the coordinates of the intersection of the diagonals of a parallelogram on a coordinate plan given the coordinates of the vertices. Example 3: a) What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with vertices M( 3, 0), N( 1, 3), P(5, 4) and R(3, 1)? b) What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with vertices L(0, 3), M( 2, 1), N(1, 5) and O(3, 1)? Example 4: Write a paragraph proof. Given: Parallelogram ABCD, AC and BD are diagonals, and point P is the intersection of AC and BD. Prove: AC and BD bisect each other. 79

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11 Geometry Section 6.2 Worksheet Name: For numbers 1 4, find the value of each variable For numbers 5 8, use parallelogram RSTU to find the measure or value. 5. m RST 6. m STU 7. m TUR 8. b For numbers 9 & 10, find the coordinates of the intersection of the diagonals of parallelogram PRYZ with the given vertices. 9. P(2, 5), R(3, 3), Y( 2, 3), Z( 3, 1) 10. P(2, 3), R(1, 2), Y( 5, 7), Z( 4, 2) 11. Write a paragraph proof of the following. Given: parallelogram PRST and parallelogram PQVU Prove: V S 12. Mr. Rodriquez used the parallelogram at the right to design a herringbone pattern for a paving stone. He will use the paving stone for a sidewalk. If m 1 is 130, find m 2, m 3, and m 4. 81

12 13. A walkway is made by adjoining four parallelograms as shown. Are the end segments a and e parallel to each other? Explain. 14. Four friends live at the four corners of a block shaped like a parallelogram. Gracie lives 3 miles away from Kenny. How far apart do Teresa and Travis live from each other? 15. Four soccer players are located at the corners of a parallelogram. Two of the players in opposite corners are the goalies. In order for goalie A to be able to see the three others, she must be able to see a certain angle x in her field of vision. What angle does the other goalie have to be able to see in order to keep an eye on the other three players? 16. Make a Venn diagram showing the relationship between squares, rectangles, and parallelograms. 17. On vacation, Tony s family took a helicopter tour of the city. The pilot said the newest building in the city was the building with this top view. He told Tony that the exterior angle by the front entrance is 72. Tony wanted to know more about the building, so he drew this diagram and used his geometry skills to learn a few more things. The front entrance is next to vertex B. a) What are the measures of the four angles of the parallelogram? b) How many pairs of congruent triangles are there in the figure? What are they? 82

13 Geometry Section 6.3 Notes: Tests for Parallelograms If a quadrilateral has each pair of opposite sides parallel, it is a parallelogram by definition. This is not the only test, however, that can be used to determine if a quadrilateral is a parallelogram. Example 1: a) Determine whether the quadrilateral is a parallelogram. Justify your answer. b) Which theorem would prove the quadrilateral is a parallelogram? 83

14 You can use the conditions of parallelograms to prove relationships in real-world situations. Example 2: Scissor lifts, like the platform lift shown, are commonly applied to tools intended to lift heavy items. In the diagram, A C and B D. Explain why the consecutive angles will always be supplementary, regardless of the height of the platform. You can also use the conditions of parallelograms along with algebra to find missing values that make a quadrilateral a parallelogram. Example 3: a) Find x and y so that the quadrilateral is a parallelogram. AB because.. AB = because.. 4x 1 = because. b) Find m so that the quadrilateral is a parallelogram. 84

15 We can use the Distance, Slope, and Midpoint Formulas to determine whether a quadrilateral in the coordinate plane is a parallelogram. Example 4: Quadrilateral QRST has vertices Q( 1, 3), R(3, 1), S(2, 3), and T( 2, 1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula. Example 5: Write a coordinate proof for the following statement: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. 85

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17 Geometry Section 6.3 Worksheet Name: For numbers 1 4, determine whether each quadrilateral is a parallelogram. Justify your answer For numbers 5 & 6, graph each quadrilateral with the given vertices. Determine whether the figure is a parallelogram. Justify your answer with the method indicated. 5. P( 5, 1), S( 2, 2), F( 1, 3), T(2, 2); Slope Formula 6. R( 2, 5), O(1, 3), M( 3, 4), Y( 6, 2); Distance and Slope Formulas For numbers 7 10, solve for x and y so that the quadrilateral is a parallelogram

18 11. The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes are parallelograms? 12. Nikia, Madison, Angela, and Shelby are balancing themselves on an X -shaped floating object. To balance themselves, they want to make themselves the vertices of a parallelogram. In order to achieve this, do all four of them have to be the same distance from the center of the object? Explain. 13. Two compass needles placed side by side on a table are both 2 inches long and point due north. Do they form the sides of a parallelogram? 14. Four jets are flying in formation. Three of the jets are shown in the graph. If the four jets are located at the vertices of a parallelogram, what are the three possible locations of the missing jet? 15. When a coordinate plane is placed over the Harrisville town map, the four street lamps in the center are located as shown. Do the four lamps form the vertices of a parallelogram? Explain. 16. Aaron is making a wooden picture frame in the shape of a parallelogram. He has two pieces of wood that are 3 feet long and two that are 4 feet long. a) If he connects the pieces of wood at their ends to each other, in what order must he connect them to make a parallelogram? b) How many different parallelograms could he make with these four lengths of wood? c) Explain something Aaron might do to specify precisely the shape of the parallelogram. 88

19 Geometry Section 6.4 Notes: Rectangles A rectangle is a parallelogram with right angles. By definition, a rectangle has the following properties. All four angles are right angles. Opposite sides are parallel and congruent. Consecutive angles are supplementary. Diagonals bisect each other. In addition, the diagonals of a rectangle are congruent. Example 1: a) A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet, and LN = 6.5 feet, find KM. LN = JN because.. JN + LN = because. LN + LN = because. 2LN = because. 2(6.5) = because.. b) Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ. You can use properties of rectangles along with algebra to find missing values. 89

20 Example 2: a) Quadrilateral RSTU is a rectangle. If m RTU = (8x + 4) and m SUR = (3x 2), solve for x. m TUR = 90 because. PT PU because.. RTU SUT because. m RTU = m SUT because. m SUT + m SUR = 90 because m RTU + m SUR = 90 because.. 8x x 2 = 90 because. b) Quadrilateral EFGH is a rectangle. If m FGE = (6x 5) and m HFE = (4x 5), solve for x. The converse of Theorem 6.13 is also true. Example 3: Some artists stretch their own canvas over wooden frames. This allows them to customize the size of canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides and the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, DA = 35 inches, BD = 37 inches, and AC = 37 inches, explain how an artist can be sure that the frame is rectangular. 90

21 You can also use the properties of rectangles to prove that a quadrilateral positioned on a coordinate plane is a rectangle given the coordinates of the vertices. Example 4: Quadrilateral JKLM has vertices J( 2, 3), K(1, 4), L(3, 2), and M(0, 3). Determine whether JKLM is a rectangle using the Distance Formula. 91

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23 Geometry Section 6.4 Worksheet Name: For numbers 1 6, quadrilateral RSTU is a rectangle. 1. If UZ = x + 21 and ZS = 3x 15, find US. 2. If RZ = 3x + 8 and ZS = 6x 28, find UZ. 3. If RT = 5x + 8 and RZ = 4x + 1, find ZT. 4. If m SUT = (3x + 6) and m RUS = (5x 4), find m SUT. 5. If m SRT = (x + 9) and m UTR = (2x 44), find m UTR. 6. If m RSU = (x + 41) and m TUS = (3x + 9), find m RSU. For numbers 7 12, quadrilateral GHJK is a rectangle. Find each measure if m 1 = m 2 8. m 3 9. m m m m 7 For numbers 13 15, graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 13. B( 4, 3), G( 2, 4), H(1, 2), L( 1, 3); Slope Formula 14. N( 4, 5), O(6, 0), P(3, 6), Q( 7, 1); Distance Formula 93

24 15. C(0, 5), D(4, 7), E(5, 4), F(1, 2); Slope Formula 16. Huntington Park officials approved a rectangular plot of land for a Japanese Zen garden. Is it sufficient to know that opposite sides of the garden plot are congruent and parallel to determine that the garden plot is rectangular? Explain. 17. Jalen makes the rectangular frame shown. In order to make sure that it is a rectangle, Jalen measures the distances BD and AC. How should these two distances compare if the frame is a rectangle? 18. A bookshelf consists of two vertical planks with five horizontal shelves. Are each of the four sections for books rectangles? Explain. 19. A landscaper is marking off the corners of a rectangular plot of land. Three of the corners are in place as shown. What are the coordinates of the fourth corner? 94

25 20. Antonio is designing a swimming pool on a coordinate grid. Is it a rectangle? Explain. 21. Veronica made the pattern shown out of 7 rectangles with four equal sides. The side length of each rectangle is written inside the rectangle. a) How many rectangles can be formed using the lines in this figure? b) If Veronica wanted to extend her pattern by adding another rectangle with 4 equal sides to make a larger rectangle, what are the possible side lengths of rectangles that she can add? 95

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27 Geometry Section 6.5 Notes: Rhombi and Squares A rhombus is a parallelogram with all four sides congruent. A rhombus has all the properties of a and the two additional characteristics described in the theorems below. Example 1: a) The diagonals of rhombus WXYZ intersect at V. If m WZX = 39.5, find m ZYX. ZX bisects WZY because.. m WZY = 2m WZX because. m QZY = 2(39.5) or 79 because.. WZ XY and ZY is a transversal because. m WZY + m ZYX = m ZYX = 180 b) The diagonals of rhombus WXYZ intersect at V. If WX = 8x 5 and WZ = 6x + 3, solve for x. 97

28 A square is a parallelogram with four sides and four angles. Recall that a parallelogram with four right angles is a rectangle, and a parallelogram with four congruent sides is a rhombus. Therefore, a parallelogram that is both a rectangle and a rhombus is also a square. The Venn diagram summarizes the relationships among parallelograms, rhombi, rectangles, and squares. All of the properties of parallelograms, rectangles, and rhombi apply to squares. For example, the diagonals of a square bisect each other (parallelogram), are congruent (rectangle), and are perpendicular (rhombus). The theorems below provide conditions for rhombi and squares. You can use properties of rhombi and squares to write proofs. 98

29 Example 2: Write a paragraph proof. Given: LMNP is a parallelogram, 1 2, and 2 6 Prove: LMNP is a rhombus. Example 3: Hector is measuring the boundary of a new garden. He wants the garden to be square. He has set each of the corner stakes 6 feet apart. What does Hector need to know to make sure that the garden is square? Example 4: Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for the given vertices: A( 2, 1), B( 1, 3), C(3, 2), and D(2, 2). List all that apply. Explain. 99

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31 Geometry Section 6.5 Worksheet Name: For numbers 1 4, PRYZ is a rhombus. If RK = 5, RY = 13 and m YRZ = 67, find each measure. 1. KY 2. PK 3. m YKZ 4. m PZR For numbers 5 8, MNPQ is a rhombus. If PQ = 3 and AP = 3, find each measure. 5. AQ 6. m APQ 7. m MNP 8. PM For numbers 9 11, use the given set of vertices to determine whether apply. Explain. BEFG is a rhombus, a rectangle, or a square. List all that 9. B( 9, 1), E(2, 3), F(12, 2), G(1, 4) 10. B(1, 3), E(7, 3), F(1, 9), G( 5, 3) 11. B( 4, 5), E(1, 5), F( 2, 1), G( 7, 1) 101

32 12. The figure is an example of a tessellation. Use a ruler or protractor to measure the shapes and then name the quadrilaterals used to form the figure. 13. A tray rack looks like a parallelogram from the side. The levels for the trays are evenly spaced. What two labeled points form a rhombus with base AA '? 14. Charles cuts a rhombus along both diagonals. He ends up with four congruent triangles. Classify these triangles as acute, obtuse, or right. 15. The edges of a window are drawn in the coordinate plane. Determine whether the window is a square or a rhombus. 16. Mackenzie cut a square along its diagonals to get four congruent right triangles. She then joined two of them along their long sides. Show that the resulting shape is a square. 17. Tatianna made the design shown. She used 32 congruent rhombi to create the flower-like design at each corner. a) What are the angles of the corner rhombi? b) What kinds of quadrilaterals are the dotted and checkered figures? 102

33 Geometry 6.6 Notes: Trapezoids and Kits A trapezoid is a quadrilateral with exactly pair of parallel sides. The sides are called bases. The nonparallel sides are called legs. The base angles are formed by the and one of the legs. In trapezoid ABCD, A and B are one pair of base angles and C and D are the other pair. If the legs of a trapezoid are congruent, then it is an isosceles trapezoid. Example 1: Each side of the basket shown is an isosceles trapezoid. If m JML = 130, KN = 6.7 feet, and LN = 3.6 feet. a) find m MJK. JK LM because. m JML + m MJK = 180 because m MJK = 180 because. m MJK = because. b) find MN. JL KM because. JL = KM because = MN because. 103

34 Example 2: Quadrilateral ABCD has vertices A(5, 1), B( 3, 1), C( 2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. The midsegment of a trapezoid is the segment that connects the of the legs of the trapezoid. The theorem below relates the midsegment and the bases of a trapezoid. Example 3: In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x? A kite is a quadrilateral with two pairs of consecutive congruent sides. Unlike a parallelogram, the sides of a kite are not congruent or parallel. 104

35 Example 4: a) If WXYZ is a kite, find m XYZ. WXY WZY because.. m WZY = because. m W + m X + m Y + m Z = because. b) If MNPQ is a kite, find NP. (NR) 2 + (MR) 2 = (MN) 2 because. c) If BCDE is a kite, find m CDE. 105

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37 Geometry Section 6.6 Worksheet Name: For numbers 1 4, find each measure. 1. m T 2. m Y 3. m Q 4. BC For numbers 5 & 6, use trapezoid FEDC, where V and Y are midpoints of the legs. 5. If FE = 18 and VY = 28, find CD. 6. If m F = 140 and m E = 125, find m D. For numbers 7 & 8, RSTU is a quadrilateral with vertices R( 3, 3), S(5, 1), T(10, 2), U( 4, 9). 7. Verify that RSTU is a trapezoid. 8. Determine whether RSTU is an isosceles trapezoid. Explain. 9. A set of stairs leading to the entrance of a building is designed in the shape of an isosceles trapezoid with the longer base at the bottom of the stairs and the shorter base at the top. If the bottom of the stairs is 21 feet wide and the top is 14 feet wide, find the width of the stairs halfway to the top. 107

38 10. A carpenter needs to replace several trapezoid-shaped desktops in a classroom. The carpenter knows the lengths of both bases of the desktop. What other measurements, if any, does the carpenter need? 11. Artists use different techniques to make things appear to be 3-dimensional when drawing in two dimensions. Kevin drew the walls of a room. In real life, all of the walls are rectangles. In what shape did he draw the side walls to make them appear 3-dimensional? 12. In order to give the feeling of spaciousness, an architect decides to make a plaza in the shape of a kite. Three of the four corners of the plaza are shown on the coordinate plane. If the fourth corner is in the first quadrant, what are its coordinates? 13. A simplified drawing of the reef runway complex at Honolulu International Airport is shown below. How many trapezoids are there in this image? 14. A light outside a room shines through the door and illuminates a trapezoidal region ABCD on the floor. Under what circumstances would trapezoid ABCD be isosceles? 15. A riser is designed to elevate a speaker. The riser consists of 4 trapezoidal sections that can be stacked one on top of the other to produce trapezoids of varying heights. All of the stages have the same height. If all four stages are used, the width of the top of the riser is10 feet. a) If only the bottom two stages are used, what is the width of the top of the resulting riser? b) What would be the width of the riser if the bottom three stages are used? 108