CHAPTER 8B GEOMETRIC PROPERTIES COMPACTED MATHEMATICS TOPICS COVERED:

Transcription

1 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.

2 Dictionary of Geometry

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4 Activity 8-16: Vocabulary Match Section 1: Polygons Word bank: Triangle Decagon Nonagon Circle Octagon Quadrilateral Hexagon Pentagon Heptagon Regular polygon Polygon 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 9. 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 Section 2: Four sided polygons (Quadrilaterals) A parallelogram with 4 right angles and 4 congruent sides 12. Word bank: A parallelogram with 4 right angles 13. (sides may or may not be congruent) Trapezoid A parallelogram with 4 congruent sides Parallelogram 14. (any size angles) Rectangle A quadrilateral with exactly one pair of opposite sides Rhombus 15. parallel (any size angles) Square A quadrilateral with opposite sides parallel and opposite 16. sides congruent 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 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 A figure that is enlarged or reduced using a scale factor dilation

5 Section 5: Angles Word bank: Angle Acute angle Right angle Straight angle Obtuse angle Vertex Diagonal An angle that is exactly An angle that is less than The point of intersection of two sides of a polygon 24. An angle that is between 90 and An angle that is exactly 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. Section 6: Figures and Angles Word bank: Congruent figures Similar figures Line of symmetry Complementary angles Supplementary angles Angles that add up to Angles that add up to Figures that are the same size and same shape 31. Figures that are the same shape and may or may not have same size Place where a figure can be folded so that both halves are congruent Section 7: Lines Word bank: Perpendicular line Ray Line Intersecting lines Parallel lines Line segment Point Plane Section 8: Triangles Word bank: Acute triangle Right triangle Obtuse triangle Scalene triangle Isosceles triangle Equilateral triangle An exact spot in space 34. A straight path that has one endpoint and extends forever in the opposite direction 35. Lines that cross at a point 36. Lines that do not cross no matter how far they are extended 37. A straight path between two endpoints 38. Lines that cross at A thin slice of space extending forever in all directions 40. A straight path that extends forever in both directions 41. A triangle with one angle of A triangle with all angles less than A triangle with no congruent sides 44. A triangle with at least 2 congruent sides 45. A triangle with an angle greater than A triangle with 3 congruent sides 47.

6 Activity 8-17: Geometry Vocabulary 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

7 A B C D E F G H I J K L M N O P Q R S T 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

8 Activity 8-18: Size and Shape 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, B E A D B E C F A C D F Corresponding side lengths are proportional. AB BC AC AB BC AB DE EF DF DE EF DF Congruent figures are the exact same shape and size. A D B E C F B E AB DE AC DF BC EF A C D F Are these triangles similar? 58 31

9 Activity 8-19: Similar Polygons Tell whether each pair of polygons is similar. 3cm cm 10 ft 6 cm 3.5 cm 3.5 cm 15 ft 8 ft 12 ft 7 cm m 100 m 4 ft 8 ft 8 ft 150 m 151 m 5. A 13 in B 13 in 12 ft E 12 in D 12 in C 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 m 12 m 15 cm x 12 cm 6 m x 20 cm cm 24 cm 10 in 35 in 30 cm x 6 in x

10 Activity 8-20: Proportions with Similar Figures 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 in 36 in 25 in 30 m x x 25 m 15 m cm 35 cm 21 cm x 14 m 20 m 60 m x mm x 25 mm 18 mm 4.5 m 6 m x 45 mm 9 m m x x 3 m 9 m 12 km 8 km 39 km 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? 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. Next, Michael measures the height of the tree, and finds it is 6 feet tall. How long is the shadow of the tree?

11 Activity 8-21: Similar Shapes Tell whether the shapes below are similar. Explain your answer 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

12 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.

13 Activity 8-22: Quadrilaterals Choose ALL, SOME, or NO 1. All Some 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. 23. I am a quadrilateral with two pairs of parallel sides and four sides of the same length. All of my angles are the same measure, too. What am I? 24. I am a quadrilateral with two pairs of parallel sides. All of my angles are 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

14 Activity 8-23: Classifying Quadrilaterals 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 Answer the following on a separate sheet of paper 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. 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?

15 Activity 8-24: Translations 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.

16 Activity 8-25: Translations Determine the coordinates of the vertices for each image of trapezoid STUW after each of the following translations in performed 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 A B C 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. 10. Connie translated trapezoid RSTU to trapezoid R S T U. 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.

17 Activity 8-26: Properties of Translations Describe the translation that maps point A to point A Draw the image of the figure after each translation 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.

18 Activity 8-27: Reflections 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)

19 Activity 8-28: Reflections 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 A B C. 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 A B C. 5. Write a symbolic representation for the reflected triangle compared to the triangle A B C. 6. Draw triangle ABC at points ( 1,1),( 3, 2),(2, 3). 7. Reflect triangle ABC across the y-axis. List the new vertices A B C. 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 A B C. 10. Write a symbolic representation for the reflected triangle compared to the triangle A B C. 11. If point Q(6,-2) is reflected across the x-axis, what will be the coordinates of point Q?

20 Activity 8-29: Properties of Reflections Use the graph for Exercises 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.

21 Activity 8-30: Rotations 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 A Three vertices B C D 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. 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. 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 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?

22 Activity 8-31: Rotations 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.

23 Activity 8-32: Properties of Rotations 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 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 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?

24 Activity 8-33: Algebraic Representations of Transformations Write an algebraic rule to describe each transformation of figure A to figure A. Then describe the transformation 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.

25 Activity 8-34: Estimating Angles Reference Angles: Determine the best estimate for each angle. Circle your answer D C A B 7. m BAC is about m CAD is about m BAD is about P O Z Q R X Y 10. m POR is about m POQ is about m QOR is about m YXZ is about m ZYX is about m YZX is about 75 40

26 Activity 8-35: Measuring Angles

27 Activity 8-36: Measuring Angles Measure Angles: Write what type of angle each is and then measure it.

28 Activity 8-37: Measuring Angles Draw the following angles using a protractor on a separate sheet of paper degree angle degree angle degree angle degree angle 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

29 Activity 8-38: Angles Complete each statement. B C A D 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. O Give the measure of each angle. U 11 RQS V T 12 RQT 13 RQU 14 RQV 15 RQW W S 16 XQW 17 XQT 18 UQV 19 VQT X Q R 20 WQS

30 Activity 8-39: Complementary/Supplementary Angles 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 Find the measure of the angle that is supplementary to the angle having the given measure Find the angle measure that is not given Name two complementary angles in the drawing at the right. 23. Name two supplementary angles in the drawing at the right. D E F A B C

31 Activity 8-40: Triangle Inequality Theorem 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, , 6, , 2, , 6, , 4, , 4, , 6, , 3, , 4, , 4, , 2, , 5, 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? m, 8 m in, 20 in cm, 9 cm ft, 7 ft cm, 3 cm mm, 13 mm

32 Activity 8-41: Angles In Triangles 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 ,30, ,30, ,60, ,46,44 Classify each triangle as equilateral, isosceles, or scalene, given the lengths of the three sides cm, 5 cm, 3 cm 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 x

33 Activity 8-42: Angles In Triangles 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 m 1 if m 2 30 and m Find m 1 if m and m Find m 4 if m 1 30 and m Find m 4 if m 1 45 and m Find m 4 if m 1 35 and m Using an equation, find x and then find the measure of the angles in each triangle x 3x 33 ( x 33) 5x x Using an equation, find x and then find the measure of the angles

34 Activity 8-43: Sides of Triangles In a triangle, the side opposite the angle with the greatest measure is the longest side.

35 Activity 8-44: Sides of Triangles 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?

36 Activity 8-45: Angles of Triangles 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 ,45 45, ,30 30, ,20 40, ,30 90, ,115 25, ,15 120, ,15 160, ,55 55, ,30 30, ,40 40, ,35 40, ,50 50, ,30 70, ,23 85,23 Explain whether the triangles are similar The diagram below shows a Howe roof truss, which is used to frame the roof of a building. 17. Explain why MPN. LQN is similar to 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.

37 Activity 8-46: Angles Use the figure below to answer questions 1 through 7. Q W 1 2 U R S V X T Describe how QR and ST are related. A. They are perpendicular lines. C. They are intersecting lines. B. They are parallel lines. D. They are complementary. Describe how WX and UV are related. A. They are perpendicular lines. C. They are intersecting lines. B. They are parallel lines. D. They are supplementary. Describe how UV and ST are related. A. They are perpendicular lines. C. They are complementary. B They are parallel lines. D. They are right angles. Which are complementary angles? A. 1 and 2 C. 3 and 4 B. 5 and 6 D. 7 and 8 Which are supplementary angles? A. 1 and 2 C. 4 and 5 B. 5 and 8 D. 7 and 8 6. If the measure of 5 is 45, what is the measure of 6? 7. What is the measure of 3? 8. What is the measure of x in the parallelogram? 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 x 40 x 10 3x x 120 x 100 x 12. 3x 2x 5x

38 Activity 8-47: Angles in Polygons Find the value of x x 75 x x x 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. 9. 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 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 13. octagon gon gon gon gon 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

39 Activity 8-48: Angles in Lines 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.

40 Activity 8-49: Angles in Lines 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 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 If m 2 57, find m 3 and m If m 6 (5x 1) and m 8 (7x 23), find m 6 and m Suppose 9, which is not shown in the figure, is complementary to 4. Given that m 1 153, what is m 9?

41 Activity 8-50: Parallel Lines Cut by a Transversal Use the figure below for Exercises Name both pairs of alternate interior angles. 2. Name the corresponding angle to Name the relationship between 1 and Name the relationship between 2 and Name the interior angles that is supplementary to Name the exterior angles that is supplementary to 5. Use the figure at the right for problems Line MP line QS. Find the angle measures. 7. m KRQ when m KNM m QRN when m MNR 52 If m RNP (8x 63) and m NRS 5x, find the following angle measures. 9. m RNP 10. m NRS In the figure at the right, there are no parallel lines. Use the figure for problems Name both pairs of alternate exterior angles. 12. Name the corresponding angle to Name the relationship between 3 and Are there any supplementary angles? If so, name two pairs. If not, explain why not.

42 Activity 8-51: Angles in Lines 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?

43 Activity 8-52: Angle Relationships/Parallel Lines Find the value of x in each figure Each pair of angles is either complementary or supplementary. Find the measure of each angle ( x 20) In the figure at the right m n. If the measure of 3 is 95, find the measure of each angle In the figure at the right m n. Find the measure of each angle m n

44 Activity 8-53: Angles Figure 1 Figure 2 A C A C B B E D E D 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 m 1 120, find m If m 3 110, find m If m 2 13, find m If m 4 65, find m 1. Find the value of x in each figure

45 Activity 8-54: Names of Polygons The word gon is derived from the Greek word gonu. Gonu means knee, which transferred to the word angle in English. SIDES NAME SIDES NAME 1 monogon 21 icosikaihenagon 2 digon 22 icosikaidigon 3 trigon or triangle 23 icosikaitrigon 4 tetragon or quadrilateral 24 icosikaitetragon 5 pentagon 25 icosikaipentagon 6 hexagon 26 icosikaihexagon 7 heptagon or septagon 27 icosikaiheptagon 8 octagon 28 icosikaioctagon 9 enneagon or nonagon 29 icosikaienneagon 10 decagon 30 triacontagon 11 hendecagon 31 tricontakaihenagon 12 dodecagon 40 tetracontagon 13 triskaidecagon 41 tetracontakaihenagon 14 tetrakaidecagon or tetradecagon 50 pentacontagon 15 pentakaidecagon or pentadecagon 60 hexacontagon 16 hexakaidecagon or hexadecagon 70 heptacontagon 17 heptakaidecagon 80 octacontagon 18 octakaidecagon 90 enneacontagon 19 enneakaidecagon 100 hectogon or hecatontagon 20 icosagon 1000 chiliagon myriagon

46 Activity 8-55: Photographs 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.