To identify congruence transformations To prove triangle congruence using isometries

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1 9-5 ongruence ransformations ommon ore tate tandards G-.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent... lso G-.B.6, G-.B.8 M 1, M 3, M bjective o identif congruence transformations o prove triangle congruence using isometries re there other methods ou could use to create two identical wings? uppose that ou want to create two identical wings for a model airplane. You draw one wing on a large sheet of tracing paper, fold it along the dashed line, and then trace the first wing. How do ou know that the two wings are identical? esson ocabular congruent congruence transformation MHMI I In the olve It, ou ma have used the properties of rigid motions to describe wh the wings are identical. ssential Understanding You can use compositions of rigid motions to understand congruence. roblem 1 Identifing qual Measures How can ou use the properties of isometries to find equal angle measures and equal side lengths? Isometries preserve angle measure and distance, so identif corresponding angles and corresponding side lengths. he composition ( n r (90, ) )(M) = GHK is shown at the right. Which angle pairs have equal measures? Because compositions of isometries preserve angle measure, corresponding angles have equal measures. m = m G, m M = m H, m = m, and m = m K. B Which sides have equal lengths? B definition, isometries preserve distance. o, corresponding side lengths have equal measures. M = GH, M = H, = K, and = GK. H G K n M 1. he composition ( t 6, 37 )( B) = XYZ. ist all of the pairs of angles and sides with equal measures. 578 hapter 9 ransformations

2 In roblem 1 ou saw that compositions of rigid motions preserve corresponding side lengths and angle measures. his suggests another wa to define congruence. Ke oncept ongruent igures wo figures are congruent if and onl if there is a sequence of one or more rigid motions that maps one figure onto the other. Does one rigid motion count as a sequence? Yes. It is a sequence of length 1. roblem Identifing ongruent igures Which pairs of figures in the grid are congruent? or each pair, what is a sequence of rigid motions that maps one figure to the other? igures are congruent if and onl if there is a sequence of rigid motions that maps one figure to the other. o, to find congruent figures, look for sequences of translations, rotations, and reflections that map one figure to another. Because r (180, ) ( D) = M, the triangles are congruent. Because ( 6-1, )(B) = WXYZ, the trapezoids are congruent. Because 6-, 97 (HG) = Q, the line segments are congruent. 6 Q X W D Y Z 6 6 B G 6 M H. Which pairs of figures in the grid are congruent? or each pair, what is a sequence of rigid motions that map one figure to the other? 6 D U B W 6 M 6 H K Q I 6 esson 9-5 ongruence ransformations 579

3 Because compositions of rigid motions take figures to congruent figures, the are also called congruence transformations. roblem 3 Identifing ongruence ransformations In the diagram at the right, W. What is a congruence transformation that maps Q onto W? he coordinates of the vertices of the triangles W sequence of rigid motions that maps Q onto W Identif the corresponding parts and find a congruence transformation that maps the preimage to the image. hen use the vertices to verif the congruence transformation. Q Because W lies above Q on the plane, a translation can map Q up on the plane. lso, notice that W is on the opposite side of the -ais and has the opposite orientation of Q. his suggests that the triangle is reflected across the -ais. It appears that a translation of Q up 5 units, followed b a reflection across the -ais maps Q to W. erif b using the coordinates of the vertices. 60, 57 (, ) = (, + 5) 60, 57 () = (, (, ) = (-, ) = et, verif that the sequence maps Q to W and to. 60, 57 (Q) = (1, 1) 60, 57 () = (5, (1, 1) = (-1, 1) = (5, ) = (-5, ) = o, the congruence 60, 57 maps Q onto W. ote that there are other possible congruence transformations that map Q onto W. W Q 3. What is a congruence transformation that maps to BY? Y B 580 hapter 9 ransformations

4 In hapter, ou studied triangle congruence postulates and theorems. You can use congruence transformations to justif criteria for determining triangle congruence. roof roblem erifing the ostulate How do ou show that the two triangles are congruent? ind a congruence transformation that maps one onto the other. Given:,, rove: tep 1 ranslate so that points and coincide. tep Because, ou can rotate about point so that and coincide. tep 3 eflect across. Because reflections preserve angle measure and distance, and because and, ou know that the reflection maps to and to. ince points and coincide, coincides with. here is a congruence transformation that maps onto, so.. erif the postulate. Given: D, Y, YD rove: YD Y D In roblem, ou used the transformational approach to prove triangle congruence. Because this approach is more general, ou can use what ou know about congruence transformations to determine whether an two figures are congruent. esson 9-5 ongruence ransformations 581

5 roblem 5 Determining ongruence How can ou determine whether the figures are congruent? You can find a congruence transformation that maps igure onto igure B. Is igure congruent to igure B? plain how ou know. igure can be mapped to igure B b a sequence of reflections or a simple translation. o, igure is congruent to igure B because there is a congruence transformation that maps one to the other. 5. re the figures shown at the right congruent? plain. igure igure B esson heck Do ou know HW? Use the graph for ercises 1 and. 1. Identif a pair of congruent figures and write a congruence statement.. What is a congruence transformation that relates two congruent figures? 6 B K 6 I Q Do ou UDD? MHMI I 3. How can the definition of congruence in terms of rigid motions be more useful than a definition of congruence that relies on corresponding angles and sides?. easoning Is a composition of a rotation followed b a glide reflection a congruence transformation? plain. 5. pen nded What is an eample of a board game in which a game piece is moved b using a congruence transformation? ractice and roblem-olving ercises MHMI I ractice or each coordinate grid, identif a pair of congruent figures. ee roblem 1 and. hen determine a congruence transformation that maps the preimage to the congruent image D G Q B Y G D M K B I 58 hapter 9 ransformations

6 In ercises 9 11, find a congruence transformation that maps M to. ee roblem M 11. M M 1. erif the ostulate for triangle congruence b using congruence roof transformations. Given: K H rove: K H ee roblem. K H K 13. erif the ostulate for triangle congruence b using congruence roof transformations. H Given: I rove: Z IQ I Q Z Q Z In ercises 1 16, determine whether the figures are congruent. If so, describe a congruence transformation that maps one to the other. If not, eplain. ee roblem B ppl onstruction he figure at the right shows a roof truss of a new building. Identif an isometr or composition of isometries to justif each of the following statements. 17. riangle 1 is congruent to triangle riangle 1 is congruent to triangle riangle is congruent to triangle 5. esson 9-5 ongruence ransformations 583

7 0. ocabular If two figures are, then there is an isometr that maps one figure onto the other. 1. hink bout a lan he figure at the right shows two congruent, isosceles triangles. What are four different isometries that map the top triangle onto the bottom triangle? How can ou use the three basic rigid motions to map the top triangle onto the bottom triangle? What other isometries can ou use?. Graphic Design Most companies have a logo that is used on compan letterhead and signs. graphic designer sketches the logo at the right. What congruence transformations might she have used to draw this logo? 3. rt rtists frequentl use congruence transformations in their work. he artworks shown below are called tessellations. What tpes of congruence transformations can ou identif in the tessellations? a. b.. In the footprints shown below, what congruence transformations can ou use to etend the footsteps? 5. rove the statements in parts (a) and (b) to show congruence in terms of roof transformations is equivalent to the criteria for triangle congruence ou learned in hapter. a. If there is a congruence transformation that maps B to D then corresponding pairs of sides and corresponding pairs of angles are congruent. b. In B and D, if corresponding pairs of sides and corresponding pairs of angles are congruent, then there is a congruence transformation that maps B to D. 58 hapter 9 ransformations

8 6. Baking ookie makers often use a cookie press so that the cookies all look the same. he baker fills a cookie sheet for baking in the pattern shown. What tpes of congruence transformations are being used to set each cookie on the sheet? hallenge 7. Use congruence transformations to prove the Isosceles riangle heorem. roof Given: G H H rove: G H 8. easoning You project an image for viewing in a large classroom. Is the projection of the image an eample of a congruence transformation? plain our reasoning. G tandardized est rep / 9. o the nearest hundredth, what is the value of in the diagram at the right? 30. In GH and XYZ, G and Y are right angles. H XZ and GH YZ. If GH = 7 ft and XY = 9 ft, what is the area of GH in square inches? 31. B is isosceles with base B. oint D is on B and D is the bisector of. If D = 5 in. and DB = in., what is B to the nearest tenth of an inch? 0 3. wo angle measures of K are 30 and 60. he shortest side measures 10 cm. What is the length, in centimeters, of the longest side of the triangle? Mied eview 33. triangle has vertices (3, ), B(, 1), and (, 3). ind the coordinates of the images of, B, and for a glide reflection with translation (, ) (, + 1) and reflection line = 0. he lengths of two sides of a triangle are given. What are the possible lengths for the third side? ee esson 9-. ee esson in., 6 in ft, 0.5 ft m, 9 m d, 8 d Get ead! o prepare for essons 9-6, do ercises Determine the scale drawing dimensions of each room using a scale of ee esson in. = 1 ft. 38. kitchen: 1 ft b 16 ft 39. bedroom: 8 ft b 10 ft 0. laundr room: 6 ft b 9 ft esson 9-5 ongruence ransformations 585