3.2 Proving Figures are Congruent Using Rigid Motions

Transcription

1 Name lass ate 3.2 Proving igures are ongruent Using igid otions ssential uestion: How can ou determine whether two figures are congruent? esource ocker plore onfirming ongruence Two plane figures are congruent if and onl if one can be obtained from the other b a sequence of rigid motions (that is, b a sequence of reflections, translations, and/or rotations). landscape architect uses a grid to design the landscape around a mall. Use tracing paper to confirm that the landscape elements are congruent. Trace planter. escribe a transformation ou can use to move the tracing paper so that planter is mapped onto planter GH. What does this confirm about the planters? H G Trace pools and NP. old the paper so that pool is mapped onto pool NP. escribe the transformation. What does this confirm about the pools? P Houghton ifflin Harcourt Publishing ompan etermine whether the lawns are congruent. Is there a rigid transformation that maps N to? What does this confirm about the lawns? eflect N N 1. How do the sizes of the pairs of figures help determine if the are congruent? odule esson 2

2 plain 1 etermining if igures are ongruent ample 1 Use the definition of congruence to decide whether the two figures are congruent. plain our answer. The two figures appear to be the same size and shape, so look for a rigid transformation that will map one to the other ou can map onto b reflecting over the -ais. This reflection is a rigid motion that maps to, so the two figures are congruent. - The coordinate notation for the reflection is (, ) (-, ). - The two figures appear to be the same/different ou can map to b. This is/is not a rigid motion that maps to, so the two figures are/are not congruent. - The coordinate notation for the rotation is. our Turn Use the definition of congruence to decide whether the two figures are congruent. plain our answer W Houghton ifflin Harcourt Publishing ompan odule 3 12 esson 2

3 plain 2 inding a Sequence of igid otions The definition of congruence tells ou that when two figures are known to be congruent, there must be some sequence of rigid motions that maps one to the other. ample 2 The figures shown are congruent. ind a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations ou use. P P ap to P with a rotation of 10 around the origin, followed b a horizontal translation. otation: (, ) (-, -) Translation: (, ) ( + 1, ) ap to with a followed b a. : (, ) : (, ), eflect. How is the orientation of the figure affected b a sequence of transformations? Houghton ifflin Harcourt Publishing ompan our Turn The figures shown are congruent. ind a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations ou use. 5. W 6. PST S0 W - T - P odule esson 2

4 plain 3 Investigating ongruent Segments and ngles ongruence can refer to parts of figures as well as whole figures. Two angles are congruent if and onl if one can be obtained from the other b rigid motions (that is, b a sequence of reflections, translations, and/or rotations.) The same conditions are required for two segments to be congruent to each other. ample 3 etermine which angles or segments are congruent. escribe transformations that can be used to verif congruence. and are congruent. The transformation is a translation. There is no transformation that maps to either of the other angles. _ and are congruent. sequence of transformations is a and a translation. There is no transformation that maps to either of the other segments. our Turn 7. etermine which segments and which angles are congruent. escribe transformations that can be used to show the congruence. G H laborate. an ou sa two angles are congruent if the have the same measure but the segments that identif the ras that form the angle are different lengths? 9. iscussion an figures have congruent angles but not be congruent figures? Houghton ifflin Harcourt Publishing ompan 10. ssential uestion heck-in an ou use transformations to prove that two figures are not congruent? odule esson 2

5 valuate: Homework and Practice Use the definition of congruence to decide whether the two figures are congruent. plain our answer. Give coordinate notation for the transformations ou use. Online Homework Hints and Help tra Practice W G 3.. P T S N - Houghton ifflin Harcourt Publishing ompan 5. P S odule esson 2

6 The figures shown are congruent. ind a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations ou use. 6. STU W 7. T S - U W GH PST 9. W G H W S T - P - etermine which of the angles are congruent. Which transformations can be used to verif the congruence? Houghton ifflin Harcourt Publishing ompan odule esson 2

7 etermine which of the segments are congruent. Which transformations can be used to verif the congruence? Use the definition of congruence to decide whether the two figures are congruent. plain our answer. Give coordinate notation for the transformations ou use. 1. W N Houghton ifflin Harcourt Publishing ompan 16. U 17. S T - 0 G H - W 0 - odule esson 2

8 The figures shown are congruent. ind a sequence of transformations for the indicated mapping. Give coordinate notation for the transformations ou use. 1. ap PST to GH. 19. ap W to. T S P W 0 - G H ap PSTU to. 21. ap to. - 0 S P T U etermine whether each pair of angles is congruent or not congruent. Select the correct answer for each lettered part. a. and ongruent Not congruent b. and ongruent Not congruent c. and ongruent Not congruent d. and ongruent Not congruent e. and ongruent Not congruent Houghton ifflin Harcourt Publishing ompan odule 3 13 esson 2

9 23. If and W are congruent, then can be mapped to W using a rotation and a translation. etermine whether the statement is true or false. Then eplain our reasoning. W 2. Which segments are congruent? Which are not congruent? plain. 25. Which angles are congruent? Which are not congruent? plain. G H 26. The figures shown are congruent. ind a sequence of transformations that will map G to STU. Give coordinate notation for the transformations ou use. 27. The figures shown are congruent. ind a sequence of transformations that will map N to. Give coordinate notation for the transformations ou use. Houghton ifflin Harcourt Publishing ompan G - -U 0 T S Which sequence of transformations does not map a figure onto a congruent figure? plain.. otation of 10 about the origin, reflection across the -ais, horizontal translation (, ) ( +, ). eflection across the -ais, combined translation (, ) ( -5, + 2). otation of 10 about the origin, reflection across the -ais, dilation (, ) (2, 2). ounterclockwise rotation of 90 about the origin, reflection across the -ais, combined translation (, ) ( -11, - 12) 0 - N odule esson 2

10 29. The figures shown are congruent. ind a sequence of transformations that will map GH to VW. Give coordinate notation for the transformations ou use. H G - - W 0 V 30. How can ou prove that two arrows in the reccling smbol are congruent to each other? 31. The cit of St. ouis was settled b the rench in the mid 1700s and joined the United States in 103 as part of the ouisiana Purchase. The cit flag reflects its rench histor b featuring the fleur-de-lis. How can ou prove that the left and right petals are congruent to each other? 32. raw onclusions Two students are tring to show that the two figures are congruent. The first student decides to map G to PST using a rotation of 10 around the origin, followed b the translation (, ) (, + 6). The second student believes the correct transformations are a reflection across the -ais, followed b the vertical translation (, ) (, - 2 ). re both students correct, is onl one student correct, or is neither student correct? T S P G - Houghton ifflin Harcourt Publishing ompan Image redits: Imagelub/Gett Images; tlaspi/shutterstock - odule esson 2

11 33. ustif easoning Two students are tring to show that the two figures are congruent. The first student decides to map G to STU using a rotation of 10 about the origin, followed b the vertical translation (, ) (, + ). The second student uses a reflection across the -ais, followed b the vertical translation (, ) (, + ), followed b a reflection across the -ais. re both students correct, is onl one student correct, or is neither student correct? S T G U H.O.T. ocus on Higher Order Thinking 3. ook for a Pattern ssume the pattern of congruent squares shown in the figure continues forever. Write rules for rigid motions that map square 0 0 onto square 1, square 0 onto square 2, and square 0 onto square Write a rule for a rigid motion that maps square 0 - onto square n. Houghton ifflin Harcourt Publishing ompan 35. nalze elationships Suppose ou know that is congruent to and that is congruent to GH. an ou conclude that is congruent to GH? plain. 36. ommunicate athematical Ideas lla plotted the points (0, 0), (, 0), and (0, ). Then she drew _ and _. Give two different arguments to eplain wh the segments are congruent. odule esson 2

12 esson Performance Task The illustration shows how nine congruent shapes can be fitted together to form a larger shape. ach of the shapes can be formed from Shape #1 through a combination of translations, reflections, and/or rotations escribe how each of Shapes 2 9 can be formed from Shape #1 through a combination of translations, reflections, and/or rotations. Then design a figure like this one, using at least eight congruent shapes. Number the shapes. Then describe how each of them can be formed from Shape #1 through a combination of translations, reflections, and/or rotations. Houghton ifflin Harcourt Publishing ompan odule 3 13 esson 2