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COMPACTED MATHEMATICS CHAPTER 8B GEOMETRIC PROPERTIES TOPICS COVERED: Geometry vocabulary Similarity and congruence Classifying quadrilaterals Transformations (translations, reflections, rotations) Measuring angles Complementary and supplementary angles Triangles (sides, angles, and side-angle relationships) Angle relationships with transversals Geometry is the area of mathematics that deals with the properties of points, lines, surfaces, and solids. It is derived from the Greek “geometra” which literally means earth measurement. Dictionary of Geometry Name: Activity 8-16: Vocabulary Match Name: Section 1: Polygons A geometric figure with 3 or more sides and angles 1. A polygon with 3 sides 2. A polygon with 4 sides 3. A polygon with 5 sides 4. A polygon with 6 sides 5. A polygon with 7 sides 6. A polygon with 8 sides 7. A polygon with 9 sides 8. A polygon with 10 sides The set of all points in a plane that are the same distance from a given point (hint: not a polygon) A polygon with all sides congruent and all angles congruent 9. Word bank: Triangle Decagon Nonagon Circle Octagon Quadrilateral Hexagon Pentagon Heptagon Regular polygon Polygon Section 2: Four sided polygons (Quadrilaterals) A parallelogram with 4 right angles and 4 congruent sides Word bank: A parallelogram with 4 right angles (sides may or may not be congruent) Trapezoid A parallelogram with 4 congruent sides Parallelogram (any size angles) Rectangle A quadrilateral with exactly one pair of opposite sides Rhombus parallel (any size angles) Square A quadrilateral with opposite sides parallel and opposite sides congruent 10. 11. 12. 13. 14. 15. 16. Section 3: Shape movement Word bank: Transformation Reflection Rotation Translation Dilation Any kind of movement of a geometric figure 17. A figures that slides from one location to another without changing its size or shape 18. A figure that is turned without changing its size or shape 19. A figure that is flipped over a line without changing its size or shape 20. A figure that is enlarged or reduced using a scale factor 21. dilation Section 5: Angles An angle that is exactly 180 22. An angle that is less than 90 23. The point of intersection of two sides of a polygon 24. An angle that is between 90 and 180 25. An angle that is exactly 90 26. A segment that joins two vertices of a polygon but is not a side 27. A figure formed by two rays that begin at the same point 28. Word bank: Angle Acute angle Right angle Straight angle Obtuse angle Vertex Diagonal Section 6: Figures and Angles Word bank: Angles that add up to 90 29. Angles that add up to 180 30. Congruent figures Figures that are the same size and same shape 31. Similar figures Figures that are the same shape and may or may not 32. Line of symmetry have same size Complementary angles Place where a figure can be folded so that both Supplementary angles 33. halves are congruent Section 7: Lines Word bank: Perpendicular line Ray Line Intersecting lines Parallel lines Line segment Point Plane An exact spot in space A straight path that has one endpoint and extends forever in the opposite direction Lines that cross at a point Lines that do not cross no matter how far they are extended A straight path between two endpoints 34. Lines that cross at 90 39. A thin slice of space extending forever in all directions 40. A straight path that extends forever in both directions 41. 35. 36. 37. 38. Section 8: Triangles Word bank: Acute triangle Right triangle Obtuse triangle Scalene triangle Isosceles triangle Equilateral triangle A triangle with one angle of 90 42. A triangle with all angles less than 90 43. A triangle with no congruent sides 44. A triangle with at least 2 congruent sides 45. A triangle with an angle greater than 90 46. A triangle with 3 congruent sides 47. Activity 8-17: Geometry Vocabulary Name: Polygons Triangles Regular polygon Equilateral triangles Quadrilaterals Scalene triangles Pentagons Isosceles triangles Hexagons Acute triangles Heptagons Right triangles Octagons Obtuse triangles Nonagons Rectangles Decagons Squares Circles Parallelograms Ovals Rhombuses Lines Trapezoids Rays Line segments A B C D E F G H I J K L M N O P Q R U V W X Y Z A1 B1 C1 D1 E1 F1 G1 H1 J1 K1 L1 M1 N1 O1 P1 Q1 R1 S1 T1 S T Activity 8-18: Size and Shape Name: Figures that have the same size and shape are congruent figures. Figures that have the same shape but may be different sizes are similar figures. The symbol means “is congruent to.” The symbol means “is similar to.” Similar figures are proportional. ABC DEF Corresponding angles are congruent, E B A D B E C F A C F D Corresponding side lengths are proportional. AB BC DE EF AC AB DF DE BC AB EF DF Congruent figures are the exact same shape and size. A D B E C F B E AB DE AC DF A C D BC EF Are these triangles similar? 58 31 F Activity 8-19: Similar Polygons Name: Tell whether each pair of polygons is similar. 3cm 1. 2. 6 cm 10 ft 7 cm 3.5 cm 3.5 cm 15 ft 7 cm 8 ft 12 ft 3. 101 m 4. 100 m 4 ft 8 ft 8 ft 150 m 151 m 12 ft 5. A 13 in B 13 in 12 in E D C 12 in For each pair of similar figures write a proportion and use the proportion to find the length of x. Use a separate sheet of paper. 6. 7. x 9m 12 m 6m 15 cm 12 cm 20 cm x 8. 9. 18 cm 30 cm 24 cm x 10 in 35 in 6 in x Activity 8-20: Proportions with Similar Figures Name: For each pair of similar figures write a proportion and use the proportion to find the length of x. Use a separate sheet of paper. 1. 2. 36 in 72 in 30 m x 25 in x 25 m 3. 15 m 4. 21 cm 20 cm x x 20 m 35 cm 60 m 14 m 5. 6. 6m 4.5 m 35 mm x 25 mm 18 mm x 9m 45 mm 7. 8. x 21 m x 8 km 3m 9m 12 km 39 km 9. A flagpole casts a shadow 22 ft long. If a 4 ft tall casts a shadow 8.8 ft long at the same time of day, how tall is the flagpole? 10. A photograph is 25 cm wide and 20 cm high. It must be reduced to fit a space that is 8 cm high. Find the width of the reduced photograph. Michael wants to find the length of the shadow of a tree. He first measures a fencepost that is 3.5 feet tall and its shadow is 10.5 feet long. 11. Next, Michael measures the height of the tree, and finds it is 6 feet tall. How long is the shadow of the tree? Activity 8-21: Similar Shapes Name: Tell whether the shapes below are similar. Explain your answer. 1. 2. 3. 4. Solve 5. A rectangle made of square tiles measures 8 tiles wide and 10 tiles long. What is the length in tiles of a similar rectangle 12 tiles wide? 6. A computer monitor is a rectangle. Display A is 240 pixels by 160 pixels. Display B is 320 pixels by 200 pixels. Is Display A similar to Display B? Explain. The figures in each pair are similar. Find the unknown measures. 7. 8. 9. 10. AMBIGRAMS A graphic artist named John Langdon began to experiment in the 1970s with a special way to write words as ambigrams. Look at all the examples below and see if you can determine what an ambigram is. Activity 8-22: Quadrilaterals Choose ALL, SOME, or NO 1. All Some Name: No rectangles are parallelograms. 2. All Some No parallelograms are squares. 3. All Some No squares are rhombi. 4. All Some No rhombi are parallelograms. 5. All Some No trapezoids are rectangles. 6. All Some No quadrilaterals are squares. 7. All Some No rhombi are squares. 8. All Some No parallelograms are trapezoids. 9. All Some No rectangles are rhombi. 10. All Some No squares are rectangles. 11. All Some No rectangles are squares. 12. All Some No squares are quadrilaterals. 13. All Some No quadrilaterals are rectangles. 14. All Some No parallelograms are rectangles. 15. All Some No rectangles are quadrilaterals. 16. All Some No rhombi are quadrilaterals. 17. All Some No rhombi are rectangles. 18. All Some No parallelograms are rhombi. 19. All Some No squares are parallelograms. 20. All Some No quadrilaterals are parallelograms. 21. All Some No parallelograms are quadrilaterals. 22. All Some No trapezoids are quadrilaterals. Solve each riddle. I am a quadrilateral with two pairs of parallel sides and four sides of the 23. same length. All of my angles are the same measure, too. What am I? I am a quadrilateral with two pairs of parallel sides. All of my angles are 24. the same measure, but my sides are not all the same length. What am I? 25. I am a quadrilateral with exactly one pair of parallel sides. What am I? 26. I am a quadrilateral with two pairs of parallel sides. What am I? Sketch each figure. Make your figure fit as FEW other special quadrilateral names are possible. 27. Rectangle 28. Parallelogram 29. Trapezoid 30. Rhombus Activity 8-23: Classifying Quadrilaterals Name: QUADRILATERALS (Four sided figues) Trapezoids Parallelograms Rectangles Rhombuses Squares List all the names that apply to each quadrilateral. Choose from parallelogram, rectangle, rhombus, square, and trapezoid. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Answer the following on a separate sheet of paper. 11. Evan said, “Every rectangle is a square.” Joan said, “No, you are wrong. Every square is a rectangle.” Who is right? Explain your answer on your graph paper. 12. What is the fewest number of figures you would have to draw to display a square, a rhombus, a rectangle, a parallelogram, and a trapezoid? What are the figures? 13. How are a square and a rectangle different? 14. How are a parallelogram and a rhombus different? 15. How are a square and rhombus alike? Activity 8-24: Translations Name: Consider the triangle shown on the coordinate plane. 1. Record the coordinates of the vertices of the triangle. 2. Translate the triangle down 2 units and right 5 units. Graph the translation. 3. A symbolic representation for the translated triangle would be: ( x, y ) ( x 5, y 2) 4. Write a verbal description of the translation. 5. Describe the translation above using symbolic representation. Activity 8-25: Translations Name: Determine the coordinates of the vertices for each image of trapezoid STUW after each of the following translations in performed. 1. 3 units to the left and 3 units down 2. ( x, y ) ( x, y 4) 3. ( x, y ) ( x 2, y 1) 4. ( x, y ) ( x 4, y ) 5. Find a single transformation that has the same effect as the composition of translations ( x, y ) ( x 2, y 1) followed by ( x, y ) ( x 1, y 3) . 6. Draw triangle ABC at points (1, 6), (4,1), (1,3) . 7. Draw triangle ABC is located at (5, 2), (2, 3), (7, 1) 8. Write a symbolic representation for the translated triangle compared to the original. 9. Write a description (in words) of this translation. Connie translated trapezoid RSTU to trapezoid RS T U . 10. Vertex S was at (-5,-7). If vertex S is at (-8,5), write a description of this translation. Move each vertex ___ units to the _________ and ___ units _________. Activity 8-26: Properties of Translations Name: Describe the translation that maps point A to point A′. 1. 2. _____________________________________ _____________________________________ Draw the image of the figure after each translation. 3. 3 units left and 9 units down 4. 3 units right and 6 units up 5. a. Graph rectangle J′K′L′M′, the image of rectangle JKLM, after a translation of 1 unit right and 9 units up. b. Find the area of each rectangle. _____________________________________ c. Is it possible for the area of a figure to change after it is translated? Explain. _____________________________________ _____________________________________ Activity 8-27: Reflections Name: 1. Record the coordinates of the vertices of triangle PQR. 2. Sketch the reflection of triangle PQR over the y-axis. 3. A symbolic representation for the reflected triangle would be: ( x, y ) ( x, y ) The line over which an object is reflected is called the line of reflection. 4. What is the line of reflection for the trapezoid above? 5. Write a verbal description of the translation. 6. A symbolic representation for the reflected triangle would be: ( x, y ) ( x, y ) Activity 8-28: Reflections Name: Use a piece of graph paper to draw the following. 1. Draw triangle ABC at points (4,1), (1,3), (5, 6) . 2. Reflect triangle ABC across the y-axis. List the new vertices ABC . 3. Write a symbolic representation for the reflected triangle compared to the original. 4. Reflect triangle ABC across the x-axis. List the new vertices ABC . 5. Write a symbolic representation for the reflected triangle compared to the triangle ABC . 6. Draw triangle ABC at points (1,1), (3, 2), (2, 3) . 7. Reflect triangle ABC across the y-axis. List the new vertices ABC . 8. Write a symbolic representation for the reflected triangle compared to the original. 9. Reflect triangle ABC across the x-axis. List the new vertices ABC . 10. Write a symbolic representation for the reflected triangle compared to the triangle ABC . 11. If point Q(6,-2) is reflected across the x-axis, what will be the coordinates of point Q ? Activity 8-29: Properties of Reflections Name: Use the graph for Exercises 1–3. 1. Quadrilateral J is reflected across the x-axis. What is the image of the reflection? _______________________________________________ 2. Which two quadrilaterals are reflections of each other across the y-axis? _______________________________________________ 3. How are quadrilaterals H and J related? _______________________________________________ Draw the image of the figure after each reflection. 4. across the x-axis 5. across the y-axis 6. a. Graph rectangle K′L′M′N′, the image of rectangle KLMN after a reflection across the y-axis. b. What is the perimeter of each rectangle? _____________________________________________ c. Is it possible for the perimeter of a figure to change after it is reflected? Explain. _____________________________________________ _____________________________________________ Activity 8-30: Rotations Name: A rotation is a transformation that describes the motion of a figure about a fixed point. In the table below record the vertices of each triangle. Triangle Three vertices A B C D 1. Starting at A to get to B involves a rotation of 90 clockwise about the origin. Make a conjecture about the changes in the x and y coordinates when a point is rotated clockwise 90 . 2. Starting at A to get to C involves a rotation of 180 clockwise about the origin. Make a conjecture about the changes in the x and y coordinates when a point is rotated clockwise 180 . 3. Starting at A to get to D involves a rotation of 270 clockwise about the origin. Make a conjecture about the changes in the x and y coordinates when a point is rotated clockwise 270 . 4. What would happen if A is rotated 360 clockwise about the origin? 5. What is true about the area of the triangle each time the shape is rotated? Activity 8-31: Rotations Name: Starting with the triangle above, rotate the triangle about the origin: A. 90 counterclockwise B. 180 counterclockwise C. 270 counterclockwise Record the coordinates of the original triangle. Record the coordinates of the 90 counterclockwise rotated circle. Record the coordinates of the 180 counterclockwise rotated circle. Record the coordinates of the 270 counterclockwise rotated circle. Activity 8-32: Properties of Rotations Name: Use the figures at the right for Exercises 1–5. Triangle A has been rotated about the origin. 1. Which triangle shows a 90 counterclockwise rotation? ____ 2. Which triangle shows a 180 counterclockwise rotation? ____ 3. Which triangle shows a 270 clockwise rotation? ____ 4. Which triangle shows a 270 counterclockwise rotation? ____ 5. If the sides of triangle A have lengths of 30 cm, 40 cm, and 50 cm, what are the lengths of the sides of triangle D? _____________________________________ Use the figures at the right for Exercises 6–10. Figure A is to be rotated about the origin. 6. If you rotate figure A 90 counterclockwise, what quadrant will the image be in? ____ 7. If you rotate figure A 270 counterclockwise, what quadrant will the image be in? ____ 8. If you rotate figure A 180 clockwise, what quadrant will the image be in? ____ 9. If you rotate figure A 360 clockwise, what quadrant will the image be in? ____ 10. If the measures of two angles in figure A are 60º and 120, what will the measure of those two angles be in the rotated figure? _____________________________________ Use the grid at the right for Exercises 11–12. 11. Draw a square to show a rotation of 90 clockwise about the origin of the given square in quadrant I. 12. What other transformation would result in the same image as you drew in Exercise 11? _____________________________________ Activity 8-33: Algebraic Representations of Transformations Name: Write an algebraic rule to describe each transformation of figure A to figure A′. Then describe the transformation. 1. 2. ___________________________________ ___________________________________ Use the given rule to graph the image of each figure. Then describe the transformation. 3. (x, y) (x, y) ___________________________________ 4. (x, y) (x, y) ___________________________________ Solve. 5. Triangle ABC has vertices A(2, 1), B(3, 0), and C(1, 4). Find the vertices of the image of triangle ABC after a translation of 2 units up. 6. Triangle LMN has L at (1, 1) and M at (2, 3). Triangle L′M′N′ has L′ at (1, 1) and M′ is at (3, 2). Describe the transformation. Activity 8-34: Estimating Angles Name: Reference Angles: 45 135 90 180 Determine the best estimate for each angle. Circle your answer. 35 70 1. 65 95 30 2. 120 170 3. 150 4. 85 140 D C A 50 80 5. 7. 25 110 6. 110 8. 155 9. P Q 10. 11. 12. 40 15 65 mCAD is about… 90 100 mBAD is about… 130 R Y X 160 120 40 mPOQ is about… 15 105 mQOR is about… 140 mPOR is about… 13. 14. 15. 60 35 45 mZYX is about… 25 75 mYZX is about… 40 mYXZ is about… B mBAC is about… Z O 55 25 Activity 8-35: Measuring Angles Name: Activity 8-36: Measuring Angles Name: Measure Angles: Write what type of angle each is and then measure it. Activity 8-37: Measuring Angles Name: Draw the following angles using a protractor on a separate sheet of paper. 1. 43 degree angle 2. 116 degree angle 3. 135 degree angle 4. 20 degree angle 5. 165 degree angle If you play golf, then you know the difference between a 3 iron and a 9 iron. Irons in the game of golf are numbered 1 to 10. The head of each is angled differently for different kinds of shots. The number 1 iron hits the ball farther and lower than a number 2, and so on. Use the table below to draw all the different golf club angles on the line segment below. Please use the 0 degree line as your starting point. 1 iron 15 degrees 6 iron 32 degrees 2 iron 18 degrees 7 iron 36 degrees 3 iron 21 degrees 8 iron 40 degrees 4 iron 25 degrees 9 iron 45 degrees 5 iron 28 degrees Pitching wedge 50 degrees 0 degrees 90 degrees Activity 8-38: Angles Name: Complete each statement. C B A D O 1. The figure formed by two rays from the same endpoint is an… 2. The intersection of the two sides of an angle is called its… 3. The vertex of COD in the drawing above is point… 4. The instrument used to measure angles is called a… 5. The basic unit in which angles are measured is the… 6. AOB has a measure of 90 and is called a ______ angle. 7. An angle whose measure is between 0 and 90 is an ______ angle. 8. Two acute angles in the figure are BOC and _______. 9. An angle whose measure is between 90 and 180 is an ______ angle. 10. An obtuse angle in the figure is _______. Give the measure of each angle. 11 RQS 12 RQT 13 RQU 14 RQV 15 RQW 16 XQW 17 XQT 18 UQV 19 VQT 20 WQS U T V W X S Q R Activity 8-39: Complementary/Supplementary Angles Name: Complementary angles add up to 90 . Supplementary angles add up to 180 . Find the measure of the angle that is complementary to the angle having the given measure. 1. 20 2. 67 3. 14 4. 81 5. 45 6. 74 Find the measure of the angle that is supplementary to the angle having the given measure. 7. 120 8. 56 9. 29 10. 162 11. 83 12. 1 Find the angle measure that is not given. 13. 14. 15. 47 60 16. 28 17. 35 116 72 19. 18. 20. 21. 67 73 34 22. Name two complementary angles in the drawing at the right. E D F 23. Name two supplementary angles in the drawing at the right. A B C Activity 8-40: Triangle Inequality Theorem Name: For any triangle, the sum of any two sides must be greater than the length of the third side. Can a triangle be formed using the side lengths below? If so, classify the triangle as scalene, isosceles, or equilateral. 1. 5, 5, 5 2. 1, 6, 4 3. 3, 2, 4 4. 6, 6, 4 5. 1, 4, 1 6. 4, 4, 8 7. 8, 6, 4 8. 3, 3, 7 9. 7, 4, 4 10. 8, 4, 5 11. 1, 2, 8 12. 12, 5, 13 13. Two sides of a triangle are 9 and 11 centimeters long. What is the shortest possible length in whole centimeters for the third side? 14. For the problem above what is the shortest possible length? In each of the following you are given the length of two sides of a triangle. What can you conclude about the length of the third side? 15. 10 m, 8 m 16. 14 in, 20 in 17. 6 cm, 9 cm 18. 12 ft, 7 ft 19. 11 cm, 3 cm 20. 9 mm, 13 mm Activity 8-41: Angles In Triangles Name: Acute vision = sharp vision. Acute pain = a sharp pain. An acute angle between 0 and 90 degrees has a fairly sharp vertex. Triangle Sum Theorem The sum of the three angles measures in any triangle is always equal to 180. Classify the triangles as right, acute, or obtuse, given the three angles. 1. 40,30,110 2. 60,30,90 3. 50, 60, 70 4. 90, 46, 44 Classify each triangle as equilateral, isosceles, or scalene, given the lengths of the three sides. 5. 3 cm, 5 cm, 3 cm 6. 50 m, 50 m, 50 m 7. 2 ft, 5 ft, 6 ft 8. x mm, x mm, y mm Find the value of x. Then classify each triangle as acute, right, or obtuse. 9. 10. 12. 13. 11. x 60 15. 16. 17. 14. 60 18. Activity 8-42: Angles In Triangles Name: The above triangle is equilateral. It is also an equiangular triangle since all angles are equal. Use the figure at the right to solve each of the following. 15. Find m1 if m2 30 and m3 55. 16. Find m1 if m2 110 and m3 25. 17. Find m4 if m1 30 and m2 55. 18. Find m4 if m1 45 and m2 60. 19. Find m4 if m1 35 and m2 45. 2 3 4 1 Using an equation, find x and then find the measure of the angles in each triangle. 20. 21. 22. x 33 5x ( x 33) Using an equation, find x and then find the measure of the angles. 23. 3x 24. x Activity 8-43: Sides of Triangles Name: In a triangle, the side opposite the angle with the greatest measure is the longest side. Activity 8-44: Sides of Triangles Name: Which angle is the second-largest angle? C A landscaper wants to place benches in the two larger corners of the deck below. Which corners should she choose? Activity 8-45: Angles of Triangles Name: Angle-Angle Criterion for Similarity We know that the angles of a triangle must add up to 180°. This means that if a triangle has two angle measurements of 40° and 80°, then the third angle must be 60°. Now if a second triangle has two angle measurements of 40° and 60°, we know the third angle must be 80°. This means the two triangles are the same shape, but not necessarily the same size. Alternately we may think of one as a dilation of the other. Either way we know that the triangles are similar. We call this the angle-angle criterion for similarity. Decide if the following triangles are similar and explain why using the angle-angle criterion. Triangle 1 Triangle 2 Triangle 1 Triangle 2 1. 45, 45 45, 90 2. 50, 30 30,100 3. 60, 20 40,100 4. 40, 30 90, 30 5. 25,115 25, 40 6. 5,15 120,15 7. 5,15 160,15 8. 45, 55 55, 90 9. 45, 30 30,100 10. 80, 40 40, 60 11. 105,35 40,105 12. 50, 50 50, 90 13. 80,30 70, 30 14. 72, 23 85, 23 Explain whether the triangles are similar. 15. 16. The diagram below shows a Howe roof truss, which is used to frame the roof of a building. 17. Explain why LQN is similar to MPN . 18. What is the length of support MP? Using the information given in the 19. diagram, can you determine whether LQJ is similar to KRJ ? Explain. Activity 8-46: Angles Name: Use the figure below to answer questions 1 through 7. W 1 2 U Q R S 3 4 V 1. 2. 3. 4. 5. T 6 5 7 Describe how QR and ST are related. A. They are perpendicular lines. B. They are parallel lines. Describe how WX and UV are related. A. They are perpendicular lines. B. They are parallel lines. Describe how UV and ST are related. A. They are perpendicular lines. B They are parallel lines. Which are complementary angles? A. 1 and 2 C. B. 5 and 6 D. Which are supplementary angles? A. 1 and 2 C. B. 5 and 8 D. 8 X C. They are intersecting lines. D. They are complementary. C. They are intersecting lines. D. They are supplementary. C. They are complementary. D. They are right angles. 3 and 4 7 and 8 4 and 5 7 and 8 6. If the measure of 5 is 45 , what is the measure of 6? 7. What is the measure of 3? What is the measure of x in the parallelogram? 124 56 8. x Write an equation to find x. Then find the measure of the missing angles in each triangle. Angle 1 Angle 2 Angle 3 Angle 1 Angle 2 Angle 3 9. x x 20 x 70 10. 2x 40 x 10 3x 60 11. x 120 x 100 x 12. 3x 2x 5x Activity 8-47: Angles in Polygons Name: Find the value of x. 1. 90 90 x 90 3. 2. 75 75 105 4. x 62 x 80 114 93 x 70 103 Write an equation to find x and then find all the missing angles. 5. A trapezoid with angles 115, 65, 55, and x . 6. A quadrilateral with angles 104, 60, 140, and x . 7. A parallelogram with angles 70, 110, (x+40), and x . 8. A quadrilateral with angles x, 2x, 3x, and 4x . A quadrilateral with angles ( x 30), (x-55), x, and (x 45) . Which of the following could be the angle measures in a parallelogram (all numbers are in degrees): 10. a) 19, 84, 84, 173 b) 24, 92, 92, 152 c) 33, 79, 102, 146 d) 49, 49, 131, 131 9. For any polygon with n sides, the following formula can be used to calculate the sum of the angles: 180(n 2) Find the sum of the measures of the angles of each polygon. 11. quadrilateral 12. pentagon 14. 12-gon 15. 18-gon 17. 75-gon 18. 100-gon 13. octagon 16. 30-gon For any polygon with n sides, the following formula can be used to calculate the average angle of size: 180( n 2) n Find the measure of each angle of each regular polygon (nearest tenth). 19. regular octagon 20. regular pentagon 21. regular heptagon 22. regular nonagon 23. regular 18-gon 24. regular 25-gon Activity 8-48: Angles in Lines Name: A transversal is a line that intersects two or more other lines to form eight or more angles. 1. Name three pairs of angles above that are supplementary. 2. Which angles appear to be acute? 3. Which angles appear to be obtuse? 4. If 1 ( x 25) and 2 85 find the size of all the other listed angles. 5. If 1 5x and 2 65 find the size of all the other listed angles. Alternate angles are on opposite sides of the transversal and have a different vertex. There are two pairs of angles in the diagram that are referred to as alternate exterior angles and two pairs of angles that are referred to as alternate interior angles. Activity 8-49: Angles in Lines Name: Adjacent angles have a common side and vertex but no common interior points. Vertical angles are pairs of nonadjacent angles formed when two lines intersect. They share a vertex but have no common rays. 1 8 2 7 3 6 4 5 1. Name all pairs of vertical angles in the figure. 2. Name all pairs of alternate interior angles in the figure. 3 Name all pairs of alternate exterior angles in the figure. 4. Name all pairs of corresponding angles in the figure. 5. Name two pairs of adjacent angles. 6. Name all of the angles that are supplementary to 8. 7. If m2 57, find m3 and m4. 8. If m6 (5 x 1) and m8 (7 x 23), find m6 and m8. 9. Suppose 9 , which is not shown in the figure, is complementary to 4 . Given that m1 153, what is m9? Activity 8-50: Parallel Lines Cut by a Transversal Name: Use the figure below for Exercises 1–6. 1. Name both pairs of alternate interior angles. 2. Name the corresponding angle to 3 . 3. Name the relationship between 1 and 5 . 4. Name the relationship between 2 and 3 . 5. Name the interior angles that is supplementary to 7 . 6. Name the exterior angles that is supplementary to 5 . Use the figure at the right for problems 7–10. Line MP || line QS. Find the angle measures. 7. mKRQ when mKNM 146 ____ 8. mQRN when mMNR 52 ____ If mRNP (8x 63) and mNRS 5x, find the following angle measures. 9. mRNP _________________ 10. mNRS _________________ In the figure at the right, there are no parallel lines. Use the figure for problems 11–14. 11. Name both pairs of alternate exterior angles. ________________________________________ 12. Name the corresponding angle to 4 ____ 13. Name the relationship between 3 and 6. ________________________________________ 14. Are there any supplementary angles? If so, name two pairs. If not, explain why not. Activity 8-51: Angles in Lines Name: An exterior angle of a triangle is formed by extending a side of the triangle. Describe the relationship between ZYX and XYM . Find the measure of each of the three exterior angles. For each exterior angle of a triangle, the two nonadjacent interior angles are its remote interior angles. Complete the table below. Exterior Angle Size of each remote interior Sum of the two remote Exterior Angle Size angle interior angles A B C Use the diagram at the right to answer each question below. 9. What is the measure of DEF? ________________________________________ 10. What is the measure of DEG? ________________________________________ Activity 8-52: Angle Relationships/Parallel Lines Name: Find the value of x in each figure. 1. 2. 3. Each pair of angles is either complementary or supplementary. Find the measure of each angle. 4. 5. 6. ( x 20) In the figure at the right m n . If the measure of 3 is 95 , find the measure of each angle. 7. 1 8. 2 9. 3 10. 4 11. 5 12. 6 13. 7 14. 8 2 1 6 3 4 7 5 8 In the figure at the right m n . Find the measure of each angle. 15. 1 16. 2 17. 3 18. 4 19. 5 20. 6 21. 7 22. 8 23. 9 24. 10 8 1 2 5 6 9 m 3 4 7 10 n Activity 8-53: Angles Name: Figure 1 Figure 2 A C A B E 1. 2. C B D E Use Figure 1 to find the following: Find x. Find the measure of ABC . Find the measure of ABE . Use Figure 2 to find the following: Find x. Find the measure of ABC . Find the measure of ABE . For Exercises 1, use the figure at the right. 1. Name a pair of vertical angles. Name a pair of complementary angles. Name a pair of supplementary angles. Use the diagram to find each angle measure. 4. If m1 120 , find m3 . 5. If m3 110 , find m2 . 6. If m2 13 , find m4 . 7. If m4 65 , find m1 . Find the value of x in each figure. 8. 9. D Activity 8-54: Names of Polygons Name: The word “gon” is derived from the Greek word “gonu”. Gonu means “knee”, which transferred to the word “angle” in English. SIDES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 NAME monogon digon trigon or triangle tetragon or quadrilateral pentagon hexagon heptagon or septagon octagon enneagon or nonagon decagon hendecagon dodecagon triskaidecagon tetrakaidecagon or tetradecagon pentakaidecagon or pentadecagon hexakaidecagon or hexadecagon heptakaidecagon octakaidecagon enneakaidecagon icosagon SIDES 21 22 23 24 25 26 27 28 29 30 31 40 41 NAME icosikaihenagon icosikaidigon icosikaitrigon icosikaitetragon icosikaipentagon icosikaihexagon icosikaiheptagon icosikaioctagon icosikaienneagon triacontagon tricontakaihenagon tetracontagon tetracontakaihenagon 50 pentacontagon 60 hexacontagon 70 heptacontagon 80 90 100 1000 10000 octacontagon enneacontagon hectogon or hecatontagon chiliagon myriagon Activity 8-55: Photographs Name: Kaitie moves five pictures around on the table and then steps back to look at the arrangement. She is arranging the photographs for a page in the school yearbook. The five pictures that she is working with are all the same size square. If each photo has to share at least one side with another photo, what are all the different ways she could place the pictures on the page? (Rotations are considered the same shape. A reflection which produces a different shape is considered a new shape.) To determine all the arrangement it makes perfect since to draw out pictures. One strategy when drawing pictures is to still create them in an organized way. Otherwise, you won’t have any idea when you have completed all of your pictures. To start with the easiest is to place all 5 in a straight line. Continue drawing pictures starting with all the 4 in a straight line options and so on.
