# Unit 8 Geometry QUADRILATERALS. NAME Period

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1 Unit 8 Geometry QUADRILATERALS NAME Period 1

2 A little background Polygon is the generic term for a closed figure with any number of sides. Depending on the number, the first part of the word Poly is replaced by a prefix. The prefix used is from Greek. The Greek term for 5 is Penta, so a 5-sided figure is called a Pentagon. We can draw figures with as many sides as we want, but most of us don t remember all that Greek, so when the number is over 12, or if we are talking about a general polygon, many mathematicians call the figure an n-gon. So a figure with 46 sides would be called a 46-gon. Vocabulary Polygon - A closed plane (two-dimensional) figure made up of several line segments that are joined together. The sides do not cross each other. Exactly two sides meet at every vertex. Types of Polygons Regular - all angles are equal and all sides are the same length. Regular polygons are both equiangular and equilateral. Irregular Any polygon with any angles NOT equal and any sides NOT the same length. Equiangular - all angles are equal. Equilateral - all sides are the same length. Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is less than 180. Concave - you can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angle is more than 180. Polygon Parts Side - one of the line segments that make up the polygon. Vertex - point where two sides meet. Two or more of these points are called vertices. Diagonal - a line connecting two vertices that isn't a side. Interior Angle - Angle formed by two adjacent sides inside the polygon. Exterior Angle - Angle formed by one side of a triangle and the extension of another side. 2

3 Special Polygons Special Quadrilaterals - square, rhombus, parallelogram, rectangle, and the trapezoid. Special Triangles - right, equilateral, isosceles, scalene, acute, obtuse. Polygon Names Generally accepted names Sides n Name N-gon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 10 Decagon 12 Dodecagon Vocabulary from 3

4 NUMBER OF SIDES OF THE POLYGO N NAME OF POLYGO N NUMBER OF INTERIO R ANGLES NUMBER OF DIAGONAL S POSSIBLE FROM ONE VERTEX POINT NUMBER OF TRANGLE S FORMED FROM ONE VERTEX POINT SUM OF MEASURE S OF INTERIO R ANGLES ONE INTERIO R ANGLE MEASUR E (REGULAR POLYGON) ONE EXTERIO R ANGLE MEASUR E (REGULAR POLYGON) SUM OF EXTERIO R ANGLES MEASURE S o o n a) Compare the number of triangles to the number of sides. Do you see a pattern? b) How can you use the number of triangles formed by the diagonals to figure out the sum of all the interior angles of a polygon? c) Write an expression for the sum of the interior angles of an n-gon, using n and the patterns you found from the table. 4

5 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 n = 12 5

6 Section 8 1: Angles of Polygons Notes Date: Diagonal of a Polygon: A segment that any two vertices. Theorem 8.1: Interior Angle Sum Theorem: If a convex polygon has n sides and S is the sum of the of its interior angles, then. Example #1: Find the sum of the measures of the interior angles of the regular pentagon below. Example #2: The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. Example #3: Find the measure of each interior angle. 6

7 Theorem 8.2: Exterior Angle Sum Theorem: If a polygon is, then the sum of the measures of the angles, one at each, is 360. Example #4: Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ. 7

8 8-1 NAME DATE PERIOD Practice Angles of Polygons Find the sum of the measures of the interior angles of each convex polygon gon gon gon The measure of an interior angle of a regular polygon is given. Find the number of sides in each polygon Find the measure of each interior angle using the given information. 7. J K 8. quadrilateral RSTU with (2x 15) (3x 20) m R 6x 4, m S 2x 8 R S (x 15) x N M U T Find the measures of an interior angle and an exterior angle for each regular polygon. Round to the nearest tenth if necessary gon gon gon Find the measures of an interior angle and an exterior angle given the number of sides of each regular polygon. Round to the nearest tenth if necessary CRYSTALLOGRAPHY 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. Glencoe/McGraw-Hil l8 Glencoe Geometry

9 Date: Properties of Parallelograms Activity Step 1 Using the lines on a piece of graph paper as a guide, draw a pair of parallel lines that are at least 10 cm long and at least 6 cm apart. Using the parallel edges of your straightedge, make a parallelogram. Label your parallelogram MATH. Step 2 Look at the opposite angles. Measure the angles of parallelogram MATH. Compare a pair of opposite angles using your protractor. The opposite angles of a parallelogram are. Step 3 Two angles that share a common side in a polygon are consecutive angles. In parallelogram MATH, MAT and HTA are a pair of consecutive angles. The consecutive angles of a parallelogram are also related. Find the sum of the measures of each pair of consecutive angles in parallelogram MATH. The consecutive angles of a parallelogram are. Step 4 Next look at the opposite sides of a parallelogram. With your ruler, compare the lengths of the opposite sides of the parallelogram you made. The opposite sides of a parallelogram are. Step 5 Finally, consider the diagonals of a parallelogram. Construct the diagonals MT and HA. Label the point where the two diagonals intersect point B. Step 6 Measure MB and TB. What can you conclude about point B? Is this conclusion also true for diagonal HA? How do the diagonals relate? The diagonals of a parallelogram. 9

10 Key Concept (Parallelogram): Section 8 2: Parallelograms Notes Date: A is a quadrilateral with pairs of opposite sides. Symbols: Theorem 8.3: Opposite sides of a parallelogram are. Theorem 8.4: angles in a parallelogram are congruent. Theorem 8.5: Consecutive angles in a parallelogram are. Theorem 8.6: right angles. If a parallelogram has one angle, it has four Theorem 8.7: The of a parallelogram bisect each other. 10

11 Example #1: RSTU is a parallelogram. Find m URT, m RST, and y. Theorem 8.8: Each diagonal of a separates the parallelogram into congruent triangles. 11

12 8-2 NAME DATE PERIOD Practice Parallelograms Complete each statement about LMNP. Justify your answer. 1. L Q? L P Q M N 2. LMN? 3. LMP? 4. NPL is supplementary to?. 5. L M? ALGEBRA Use RSTU to find each measure or value. 6. m RST 7. m STU 8. m TUR 9. b R S B 23 4b 1 U T COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram PRYZ given each set of vertices. 10. P(2, 5), R(3, 3), Y( 2, 3), Z( 3, 1) 11. P(2, 3), R(1, 2), Y( 5, 7), Z( 4, 2) 12. PROOF Write a paragraph proof of the following. Given: PRST and PQVU Prove: V S T U P V Q S R 13. CONSTRUCTION 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 Glencoe/McGraw-Hill 12 Glencoe Geometry

13 Section 8 3: Tests for Parallelograms Notes Date: Conditions for a Parallelogram: By definition, the opposite sides of a parallelogram are parallel. So, if a quadrilateral has each pair of opposite sides parallel it is a parallelogram. Key Concept (Proving Parallelograms): Theorem 8.9: If both pairs of sides of a quadrilateral are, then the quadrilateral is a parallelogram. Theorem 8.10: If both pairs of opposite of a quadrilateral are, then the quadrilateral is a parallelogram. Theorem 8.11: If the of a quadrilateral each other, then the quadrilateral is a parallelogram. Theorem 8.12: If one pair of opposite sides of a quadrilateral is both and, then the quadrilateral is a parallelogram. 13

14 Example #1: Find x and y so that each quadrilateral is a parallelogram and justify your reasoning. a.) b.) 14

15 Given: VZRQ and WQST Q Prove: Z T Statements Reasons 1. VZRQ 1. Given W V R Z A T 2. Z Q 2. Opposite angles of a parallelogram are congruent 3. WQST 3. Given 4. Q T 4. Opposite angles of a parallelogram are congruent 5. Z T 5. Transitive 15

16 8-3 NAME DATE PERIOD Practice Tests for Parallelograms Determine whether each quadrilateral is a parallelogram. Justify your answer COORDINATE GEOMETRY Determine whether a figure with the given vertices is a parallelogram. Use 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 Formula ALGEBRA Find x and y so that each quadrilateral is a parallelogram. 7. (5y 9) 8. (5x 29) 4x 2 2y 8 (3y 15) (7x 11) 3y 5 3x x y 7 7y 3 4x 6 2x 6 y 23 4y 2 x TILE DESIGN The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes are parallelograms? Glencoe/McGraw-Hill 16 Glencoe Geometry

17 Date: Section 8 4: Rectangles Notes Rectangle: A quadrilateral with angles. Both of angles are. A rectangle has all the of a. Theorem 8.13: If a parallelogram is a, then the are congruent. Key Concept (Rectangle): Properties: Opposite are congruent and parallel. Opposite are. angles are supplementary. All angles are angles. 17

18 Example #1: Quadrilateral RSTU is a rectangle. If RT = 6x + 4 and SU = 7x 4, find x. Example #2: Quadrilateral LMNP is a rectangle. a.) Find x. b.) Find y. Theorem 8.14: If the of a parallelogram are congruent, then the parallelogram is a. 18

19 19

20 8-4 NAME DATE PERIOD Practice Rectangles ALGEBRA RSTU is a rectangle. 1. If UZ x 21 and ZS 3x 15, find US. R U Z S T 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 2 9 and m UTR 2x 44, find x. 6. If m RSU x 2 1 and m TUS 3x 9, find m RSU. GHJK is a rectangle. Find each measure if m m 2 8. m 3 9. m m 5 G 2 K 1 H J 11. m m 7 COORDINATE GEOMETRY Determine whether BGHL is a rectangle given each set of vertices. Justify your answer. 13. B( 4, 3), G( 2, 4), H(1, 2), L( 1, 3) 14. B( 4, 5), G(6, 0), H(3, 6), L( 7, 1) 15. B(0, 5), G(4, 7), H(5, 4), L(1, 2) 16. LANDSCAPING 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. Glencoe/McGraw-Hill 20 Glencoe Geometry

21 Properties of Rhombi and Squares: A Rhombus is a Parallelogram with all 4 sides Congruent. A square is a special type of rhombus. Rhombus Square 1. A Rhombus has all the properties 1. A square has all the properties of of a parallelogram a parallelogram 2. All sides are Congruent 2. A square has all the properties of a rectangle 3. Diagonals are Perpendicular 3. A square has all the properties of a rhombus 4. Diagonals bisect the angles of a Rhombus 21

22 Date Directions: Mark all properties that apply to each figure built. Property Both pairs of opposite sides are Figure (A) Chart the property Figure (B) Figure (C) Figure (D) Figure (E) Figure (F) Figure (G) All sides are Diagonals are Diagonals are Diagonals bisect each other Diagonals bisect pair of opposite s Both pairs of opposite s are Consecutive s are All s are Questions to answer: 1. What is the one thing that is different about the diagonals between the two types of figures created in this activity? 2. What is the one thing that is different about the angles between the two types of figures created in this activity? 22

23 Figures for activity Figure A Figure B Figure C Figure D Figure E Figure F Figure G In class activity figures A- G 23

24 Rhombus: Section 8 5: Rhombi and Squares Notes Date: A is a special type of parallelogram called a. A rhombus is a quadrilateral with all four sides. All of the of parallelograms can be applied to rhombi. Key Concept (Rhombus): Theorem 8.15: The of a rhombus are perpendicular. Theorem 8.16: If the diagonals of a are, then the parallelogram is a rhombus. Theorem 8.17: Each diagonal of a rhombus a pair of opposite. 24

25 Example #1: Use rhombus LMNP and the given information to find the value of each variable. a.) Find y if m 1 = y 54 b.) Find m PNL if m MLP = 64. Square: If a quadrilateral is both a and a, then it is a square. All of the properties of parallelograms and rectangles can be applied to. 25

26 Section 8-5 Homework date: ABCD is a rhombus. 1.) If m ABD = 60, find m BDC. 2.) If AE = 8, find AC. 3.) If AB = 26 and BD = 20, find AE. 4.) Find m CEB. 5.) If m CBD = 58, find m ACB. 6.) If AE = 3x 1 and AC = 16, find x. 7.) If m CDB = 6y and m ACB = 2 y + 10, find y. 8.) If AD = 2x + 4 and CD = 4x 4, find x. Use rhombus PRYZ with RK = 4y + 1, ZK = 7y 14, PK = 3x 1, and YK = 2x ) Find PY. 10.) Find RZ. 11.) Find RY. 12.) Find m YKZ. 26

27 8-5 NAME DATE PERIOD Practice Rhombi and Squares Use rhombus PRYZ with RK 4y 1, ZK 7y 14, PK 3x 1, and YK 2x 6. R K Y 1. Find PY. 2. Find RZ. P Z 3. Find RY. 4. Find m YKZ. Use rhombus MNPQ with PQ 3 2, PA 4x 1, and AM 9x 6. N A P 5. Find AQ. 6. Find m APQ. M Q 7. Find m MNP. 8. Find PM. COORDINATE GEOMETRY Given each set of vertices, determine whether BEFG is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning. 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( 7, 1), G( 2, 1) 12. TESSELATIONS 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. Glencoe/McGraw-Hill 27 Glencoe Geometry

28 Date: Trapezoid: Section 8 6: Trapezoids Notes A quadrilateral with exactly of parallel. The sides are called. The base are formed by the base and one of the. The sides are called. Isosceles Trapezoid: A trapezoid that has legs. Theorem 8.18: Both pairs of base of an isosceles trapezoid are. Theorem 8.19: The of an isosceles trapezoid are congruent. 28

29 Median: The segment that joins of the of a trapezoid. Theorem 8.20: The median of a trapezoid is to the bases, and its measure is the sum of the measures of the bases. Example #1: DEFG is an isosceles trapezoid with median MN. a.) Find DG if EF = 20 and MN = 30. b.) Find m 1, m 2, m 3, and m 4 if m 1 = 3x + 5 and m 3 = 6x 5. 29

30 Properties of Trapezoids: A Trapezoid is a QUADRILATERAL with exactly one pair of parallel sides. AND The parallel sides are called bases leg base leg leg base In an Isosceles Trapezoid, the legs are base congruent. AND Both pairs of base angles are congruent AND The diagonals are congruent base leg The Median of a trapezoid is a line segment that joins the midpoints of the legs. AND It is parallel to the bases and is equal to ½ the sum of the lengths of the bases. 1 EF = ( AB + 2 DC) E A base median B F D base C 30

31 8-6 NAME DATE PERIOD Practice Trapezoids COORDINATE GEOMETRY RSTU is a quadrilateral with vertices R( 3, 3), S(5, 1), T(10, 2), U( 4, 9). 1. Verify that RSTU is a trapezoid. 2. Determine whether RSTU is an isosceles trapezoid. Explain. COORDINATE GEOMETRY BGHJ is a quadrilateral with vertices B( 9, 1), G(2, 3), H(12, 2), J( 10, 6). 3. Verify that BGHJ is a trapezoid. 4. Determine whether BGHJ is an isosceles trapezoid. Explain. ALGEBRA Find the missing measure(s) for the given trapezoid. 5. For trapezoid CDEF, V and Y are 6. For trapezoid WRLP, B and C are midpoints of the legs. Find CD. midpoints of the legs. Find LP. C V F E Y D W B P L C R 7. For trapezoid FGHI, K and M are 8. For isosceles trapezoid TVZY, find the midpoints of the legs. Find FI, m F, length of the median, m T, and m Z. and m I. G 21 H T 34 V F K M I Y 5 Z 9. CONSTRUCTION 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, find the width of the stairs halfway to the top. 10. DESK TOPS 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? 60 Glencoe/McGraw-Hill 31 Glencoe Geometry

32 Date Define each type of figure draw an example special characteristics (some things to ask yourself: Diagonals =, diagonals bisect each other, diagonals bisect angle, diagonals perpendicular, angle measures equal, angle measures supplementary, sides equal, sides parallel,) 1. quadrilateral 2. parallelogram 3. square 4. rectangle 5. rhombus 6. kite 7. trapezoid 8. Isosceles trapezoid 32

33 33

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