Ready to Go On? Skills Intervention Building Blocks of Geometry

8-1 Ready to Go On? Skills Intervention Building Blocks of Geometry A point is an exact location. A line is a straight path that extends without end in opposite directions. A plane is a flat surface that extends without end in all directions. A line segment is made of two endpoints and all the points between the endpoints. A ray has one endpoint. From the endpoint, the ray extends without end in one direction only. Identifying Points, Lines, and Planes Use the diagram to name each geometric figure. Vocabulary point line plane line segment ray A. three points Name three exact locations. B. two lines Name two straight paths that extend without end in opposite I G H J K L directions. C. a point shared by two lines Name a point that is on both lines. point D. a plane Name three points that are on the plane, but not on a line.,, and Use the three points to name a plane. plane Identifying Line Segments and Rays Use the diagram to give a possible name to each figure. A. two different line segments Use the endpoints to name two different line segments. R S B. six different names for rays Q Use an endpoint first and then another point on the ray to name six rays. C. another name for ray QR What is the endpoint? What is another point on the ray? What is another name for ray QR? 156 Holt Mathematics

8-2 Ready to Go On? Skills Intervention Measuring and Classifying Angles An angle is formed by two rays with a common endpoint, called the vertex. Angles are measured in degrees. A right angle measures exactly 90 and a straight angle measures exactly 180. An acute angle measures less than 90 and an obtuse angle measures more than 90 and less than 180. Measuring an Angle with a Protractor Use a protractor to measure the angle. Tell what type of angle it is. Where will you place the center point of the protractor? Which degree mark on the protractor do you want ray Vocabulary angle vertex right angle straight angle acute angle obtuse angle A BC to pass through? Using the scale that starts with 0 along ray BC, read the measure where ray BA crosses. m ABC Is m ABC greater than, less than, or equal to 90? B C So the angle is a(n) angle. Drawing an Angle with a Protractor Use a protractor to draw an angle that measures 120. Draw a ray near the bottom of the space provided to the right. Where will you place the center point of the protractor? On the protractor, which degree mark do you want the ray to pass through? At which degree mark will you make a mark above the scale on the protractor? Where will you draw the second ray? 157 Holt Mathematics

8-2 There are 360 in a circle. You can use that fact to help solve some problems. What is the measure of the angle formed by the minute and hour hands of a clock at 1:30? Understand the Problem 1. How many degrees is it all the way around the clock face? Make a Plan Ready to Go On? Problem Solving Intervention Measuring and Classifying Angles 2. Break the angle you want to find into parts. What fraction of a whole circle is BOC? How could you use that fraction to find m BOC? A 12 D 11 1 B 10 2 9 O 8 4 7 5 6 C 3 3. If you knew m AOB, how could you find m DOB? What fraction of a whole circle would that angle be? 4. What fraction of a whole circle is AOB? How could you use that fraction to find m AOB? Solve 5. What is m BOC? Write the measure on the diagram. 6. What is m DOB? Write the measure on the diagram. 7. What is the measure of the angle formed by the hands at 1:30? Check 8. Show that your answer makes sense. Solve 9. What size angle do the hands form at 10:15? Hint: The hour hand is 1 4 of the way from 10 to 11. 158 Holt Mathematics

8-3 Ready to Go On? Skills Intervention Angle Relationships Congruent angles are angles that have the same measure. Vertical angles are formed opposite of each other when two lines intersect. Adjacent angles are side by side and have a common vertex and ray. Complementary angles are two angles whose measures have a sum of 90 and supplementary angles are two angles whose measures have a sum of 180. Identifying Types of Angle Pairs Identify the type of each angle pair shown. Vocabulary congruent vertical angles adjacent angles complementary angles supplementary angles A. Is 3 opposite 4? Are both angles formed by two intersecting lines? What type of angles are 3 and 4? 3 4 B. Are angles 5 and 6 side by side? Do they have a common vertex and ray? What type of angles are 5 and 6? 5 6 Identifying an Unknown Angle Measure Find each unknown angle measure. A. The angles are complementary. 45 A What is the sum of the measures? Subtract 45 from both sides. A Solve for A. A 45 m A B. The angles are supplementary. 35 B What is the sum of the measures? B 35 Subtract 35 from both sides. B Solve for B. m B 159 Holt Mathematics

8-4 Ready to Go On? Skills Intervention Classifying Lines Parallel lines are lines in the same plane that never intersect. Perpendicular lines intersect to form 90 angles, or right angles. Skew lines are lines that lie in different planes; they are neither parallel nor intersecting. Classifying Pairs of Lines Classify each pair of lines. Vocabulary parallel lines perpendicular lines skew lines A. B. Are the lines in the same plane? Are the lines in the same plane? Are the lines parallel? Do the lines intersect? Do the lines intersect? The lines are. The lines are. C. D. Do the lines cross at one common point? Do the lines intersect to form right angles? The lines are. The lines are. Map Application Main Street and Grand Street each run North-South. What type of line relationship does this represent? Are the streets in the same plane? Do the streets intersect? Main Street Grand Street The streets represent lines. 160 Holt Mathematics

SECTION 8A Ready to Go On? Quiz 8-1 Building Blocks of Geometry Use the diagram to name each geometric figure. 1. three points 2. two lines 3. a point shared by two lines 4. a plane 5. two line segments 6. two rays G E F H 8-2 Measuring and Classifying Angles Use a protractor to measure each angle. Then classify each angle as acute, right, obtuse, or straight. 7. 8. 9. 10. 11. The steepest slide at the new playground has a top angle of 60. Draw an angle with this measure. 161 Holt Mathematics

SECTION 8A 8-3 Angle Relationships Find each unknown angle measure. 12. 13. r Ready to Go On? Quiz continued 30 50 s 14. 15. t 24 u 20 20 8-4 Classifying Lines Classify each pair of lines. 16. 17. 18. 19. 162 Holt Mathematics

SECTION 8A Students in Mrs. Zirellis class built towers from rolling newspaper and then taping it together. In order to make a sturdy tower the supports were organized into angles. The graphic below is a sketch of one of the taller towers. Measuring and Classifying Angles Use a protractor to measure each angle. Then classify each angle as acute, right, or obtuse. 1. ABC Ready to Go On? Enrichment Paper Tower C 2. DEF B 3. GHI F D A G L E K M H 90 40 55 50 J 45 65 50 I Angle Relationships Find each unknown angle measure. 4. J 5. K 6. L 7. M 163 Holt Mathematics

8-5 Ready to Go On? Skills Intervention Triangles Triangles can be classified by the measures of their angles. An acute triangle has only acute angles. An obtuse triangle has one obtuse angle. A right triangle has one right angle. The sum of the measures of the angles in any triangle is 180. Vocabulary acute triangle obtuse triangle right triangle Acute triangle Obtuse triangle Right triangle Sports Application Hiking paths connecting lakes J, K, and L form a triangle. The measure of J is 38, and the measure of K is 44. Classify the triangle. To classify the triangle, find the measure of L. L 180 ( ) What will you subtract from 180? L 180 L The measure of L is So the hiking paths form a(n). Does JKL have an obtuse angle? triangle. Using Properties of Angles to Label Triangles Use the diagram to find the measure of GIJ. What type of angles are GIJ and FIH? So m GIJ m FIH. G m FIH 180 ( ) 180 I J m FIH so, m GIJ. F 20 25 H 164 Holt Mathematics

8-6 Ready to Go On? Skills Intervention Quadrilaterals A quadrilateral is a plane figure with four sides and four angles. There are five special types of quadrilaterals. parallelogram: opposite sides are parallel and congruent, and opposite angles are congruent rectangle: parallelogram with four right angles rhombus: parallelogram with four congruent sides square: rectangle with four congruent sides trapezoid: exactly two parallel sides, may be two right angles Vocabulary quadrilateral parallelogram rectangle rhombus square trapezoid Naming Quadrilaterals Give the most descriptive name for each figure. A. Does the figure have four sides and four angles? Are exactly two of the sides parallel? is the most exact name. B. Is this figure a plane figure? Does the figure have only four sides and four angles? The figure is a quadrilateral. C. Does the figure have four sides and four angles? Does the figure have four right angles? is the most exact name. Classifying Quadrilaterals Complete the statement. A. A rhombus can also be called a? and?. Does a rhombus have opposite sides that are parallel and congruent? So a rhombus can be called a. Is a rhombus a plane figure with four sides and four angles? So a rhombus can be called a. B. A rhombus with four right angles can also be called a?. Does a square have four congruent sides and four right angles? So a rhombus with four right angles can also be called a. 165 Holt Mathematics

8-6 Ready to Go On? Problem Solving Intervention Quadrilaterals Sometimes you can solve a problem even though you may think at first that there is not enough information. A rhombus and an equilateral triangle are adjacent, forming a trapezoid. What fraction of the perimeter of the trapezoid is the perimeter of the rhombus? Understand the Problem 1. Draw a rhombus and triangle as described in the problem. 2. What two quantities are you supposed to compare? Make a Plan 3. What do you know about length of the sides of a rhombus? Of an equilateral triangle? 4. Why must the sides of the triangle be the same length as the sides of the rhombus? 5. Mark your diagram to show which lengths are equal. Solve 6. If the length of each side of the rhombus is 1, what is the perimeter of the rhombus? The perimeter of the trapezoid? 7. What fraction of the perimeter of the trapezoid is the perimeter of the rhombus? Check 8. Does your diagram match what the problem states? Is your reasoning correct? 166 Holt Mathematics

8-7 Ready to Go On? Skills Intervention Polygons A polygon is a closed plane figure formed by three or more line segments. A regular polygon is a polygon with all sides congruent and all angles congruent. Identifying Polygons Name each polygon and tell whether it appears to be regular or not regular. Vocabulary polygon regular polygon A. How many sides are there? How many angles are there? The polygon is a(n). Do the sides and angles appear to be congruent? The polygon appears to be. B. How many sides are there? How many angles are there? The polygon is a(n). Do the sides and angles appear to be congruent? The polygon appears to be. C. How many sides are there? How many angles are there? The polygon is a(n). Do the sides and angles appear to be congruent? The polygon appears to be. Home Economics Application Jessica made a cake in the shape of a regular hexagon for her brother s sixth birthday. What is the measure of each angle of the hexagon? How many sides does a hexagon have? How many triangles are there inside a hexagon? What is the sum of the interior angles in a hexagon? 180 So the measure of each angle is or. 167 Holt Mathematics

8-8 Ready to Go On? Skills Intervention Geometric Patterns Extending Geometric Patterns Identify a possible pattern. Use the pattern to draw the next figure. What might the pattern be? So the next figure might be a. Completing Geometric Patterns Identify a possible pattern. Use the pattern to draw the missing figure. What might the pattern be? So the missing figure might be a. Art Application Shannon is drawing a picture. Identify a pattern that Shannon is using and draw what the next shape will probably be. What might the pattern be? So the missing figure might be a. 168 Holt Mathematics

SECTION 8B Ready to Go On? Quiz 8-5 Triangles Use the diagram for problems 1 and 2. 1.5 m H 1. Find m HFD. 2. Classify triangle HGE by its angles and by its sides. 3.5 m D F 5 m E 60 40 6 m G If the angles can form a triangle, classify it as acute, obtuse, or right. 3. 98, 48, 34 4. 52, 38, 90 5. 73, 57, 60 6. 75, 65, 40 8-6 Quadrilaterals Give the most descriptive name for each figure. 7. 8. 9. 10. 11. One angle of a rhombus is 73. What is the measure of the opposite angle? 12. The perimeter of a square is 168 centimeters. What is the length of one side of the square? 169 Holt Mathematics

SECTION 8B Ready to Go On? Quiz continued 8-7 Polygons Name each polygon and tell whether it appears to be regular or irregular. 13. 14. 15. 16. 8-8 Geometric Patterns Identify a possible pattern. Use the pattern to draw the missing figure. 17. 18.?? 19. 20.?? 170 Holt Mathematics

SECTION 8B Ready to Go On? Enrichment Custom Kites Here are diagrams of some kites at a local kite festival. Quadrilaterals Give all the names for each polygon. 1. 2. 3. 4. Polygons Name each polygon and tell whether it appears to be regular or irregular. 5. 6. 7. 8. 171 Holt Mathematics

8-9 Ready to Go On? Skills Intervention Congruence Identifying Congruent Figures Decide whether the figures in each pair are congruent. A. Do these figures have the same shape and size? These figures are. B. Are both of these figures squares? Do these figures have the same shape and size? These figures are. C. 6 in. Are both of these figures triangles? 8 in. 10 in. 6 in. 8 in. 10 in. Do these figures have the same shape and size? These figures are. Consumer Application Shauna needs a tablecloth that is congruent to the top of the table. Which tablecloth should she buy? Table Tablecloth A Tablecloth B 8 ft 4 ft 4 ft 2 ft 4 ft 8 ft Are both tablecloths the same shape as the table? Are both tablecloths the same size as the table? Which tablecloth is the same size and shape as the table? is congruent to the table. 172 Holt Mathematics

8-9 Ready to Go On? Problem Solving Intervention Congruence If you know that two triangles are congruent, you may be able to use that information to prove other things. In the diagram, BCE ACD. Show that B D A E. A Understand the Problem 1. Use a colored marker to outline the two congruent triangles BCE and ACD. Make a Plan 2. Mark the sides of the triangles that are congruent. 3. What two line segments make up side A C? side B C? B D C E Solve 4. Fill in each blank with the correct distance. A C A E B C A C E C A C E C B C D C 5. Use exercise 4 to explain how you know that A E B D. Check 6. Trace BCE and ACD to make sure you identified corresponding sides correctly. Solve 7. Which triangle has the larger perimeter, BCE or ACD? Explain. 8. If B C 6 cm and A E 4 cm, what is the length of D C? 173 Holt Mathematics

8-10 Ready to Go On? Skills Intervention Transformations A rigid transformation moves a figure without changing its size or shape. A translation is the movement of a figure along a straight line. A rotation is the movement of a figure around a point. When a figure flips over a line, creating a mirror image, it is called a reflection. The line the figure is flipped over is called the line of reflection. Identifying Transformations Tell whether each is a translation, rotation, or reflection. Vocabulary transformation translation rotation reflection line of reflection A. How did the figure move? The figure is a. B. How did the figure move? The figure is a. C. How did the figure move? The figure is a. Drawing Transformations Draw each transfomation. A. Draw a vertical reflection. What will you flip the figure over? Draw the figure in its new location. B. Draw a 90 clockwise rotation about the point. Where will you place your pencil? How far will you rotate the figure? Draw the figure in its new location. 174 Holt Mathematics

8-11 Ready to Go On? Skills Intervention Line Symmetry A figure has line symmetry if it can be folded or reflected so that the two parts of the figure match, or are congruent. The line of reflection is called the line of symmetry. Identifying Lines of Symmetry Determine whether the dashed line appears to be a line of symmetry. Are the two parts of the figure congruent? Do they appear to match exactly when folded or reflected across the line? The line to be a line of symmetry. Finding Multiple Lines of Symmetry A. Find all the lines of symmetry in the regular polygon. Trace the figure and cut it out. Fold the figure in half in different ways. How many different ways can you fold the figure in half? So there are lines of symmetry. B. Find all the lines of symmetry in the object. Trace the figure and cut it out. Fold the figure in half in different ways. How many different ways can you fold the figure in half? So there are lines of symmetry. Vocabulary line of symmetry line symmetry Visualizing Symmetric Figures Draw the cut-out figure Then reflect that figure as it would look unfolded. Draw a congruent figure. across the fold line. 175 Holt Mathematics

SECTION 8C Ready to Go On? Quiz 8-9 Congruence Decide whether the figures in each pair are congruent. If not, explain. 1. 2. 3. Jon needs a cover for his pool. Which cover will fit? 5 m Pool 6 m 3 m A 6 m 5 m B 5 m 5 m C 6 m 8-10 Transformations Tell whether each is a translation, rotation, or reflection. 4. 5. 6. 7. 176 Holt Mathematics

SECTION 8C Ready to Go On? Quiz continued Draw and name each transformation. 8. 9. 8-11 Line Symmetry Determine whether each dashed line appears to be a line of symmetry. 10. 11. 12. 13. 177 Holt Mathematics

SECTION 8C Ready to Go On? Enrichment Transforming Jewelry At summer camp, Jill took a jewelry-making class. She wanted to make a belt buckle similar to the ones she saw in a design book. The book instructed her to look at the placement of objects and how identical objects are put into place. Jill starts with the jeweled belt buckle. 1. The first step to assemble the belt buckle is to transform the star in the top left corner to the other three corners as shown below. Is this a translation, rotation, or reflection? 2. The next step is to transform the top center design to the lower position as shown. Is this a translation, rotation, or reflection? 3. Lastly, attach a string of stones from the center circle. Repeat several strings around the circle. Are these translations, rotations, or reflections? 4. Complete the rest of the bracelet using rotation. 5. Draw your own piece of jewelry using transformations. Label each transformation type. 178 Holt Mathematics