Lesson 1 eteach Congruence and Transformations Translation eflection otation length is the same length is the same length is the same orientation is the sarne onentation is different onentation is different \ Example Determine if the two figures are congruent by using transformations. The two triangles are congruent because a S rotation followed by a translation will map XYZ onto ST. Y Exercises Determine if the two figures are congruent by using transformations. N O 2. _ S J I Q congruent; reflection maps IJ onto ONQP. not congruent; No sequence of transformations maps PQf? onto T S U exactly. Course 3 Chapter 7 Congruence and Similarity 99
Lesson 1 Skills Practice Congruence and Transformations Determine if the two figures are congruent by using transformations. Explain your reasoning. 2. u w congruent; rotation maps Srty onto YX. congruent; translation maps EF onto JKLM. 3. T 4. D Q congruent; reflection maps T V onto UVX. not congruent; No sequence of transformations maps C D onto PQS. 5. N S 6. 0 C S P Q M N E F 0 not congruent; No sequence of transformations maps CDEF onto NOPQS. congruent; reflection maps QST onto LMNO. 100 Course 3 Chapter 7 Congruence and Similarity
Lesson 1 omework Practice Congruence and Transformations Determine if the two figures are congruent by using transformations. Explain your reasoning. 2. The two triangles are congruent because a rotation followed by a translation will map C onto ST. The two figures are congruent because a clockwise rotation of 180 followed by a translation will map the EF onto IJKL J I 4. 1 The two figures are not congruent because no sequence of transformations will map Figure onto Figure exactly. The two figures are congruent because a clockwise rotation of 180 followed by a translation will map the Figure onto Figure. 5. PIC DESIN The rt Club designed the logo shown. h a t transformations did they use if the top trapezoid is the preimage and the bottom trapezoid is the image? Sample answer: the preimage is reflected over a horizontal line 6. SCPOOKIN Charlotte used a stamp to create the pattern shown. h a t transformations did she use if parallelogram is the preimage and parallelogram is the image? Sample answer: the preimage is translated down and to the left Course 3 Chapter 7 Congruence and Similarity 99
Lesson 2 eteach Congruence If two figures are congruent, their corresponding sides are congruent and their corresponding angles are congruent. Example rite congruence statements comparing the corresponding parts in the congruent triangles shown. Use the matching arcs and tick marks to identify the corresponding parts. Corresponding angles: ZS = Z, Z = Z, ZT= ZX Corresponding S =,T sides: = X, TS = X Exercises rite congruence statements comparing the corresponding parts in the congruent figures shown. 2. P T Z s Zp,^T = ZF^S = ZE; T s DF, TS s FE, S = ED, ZU s ZJ,^ -_?!_ ^ =_Z; UP = JS, PT s S, TU s J, 3. 4. O Z ^ Z,^ =j f'j P =_j F' = F, C s FE, C ^ E, ZO s Z, zp^s!1 = Z ^ ZN; OP s fl/,p^ /S, ^ S N, O ^ N Course 3 Chapter 7 Congruence and Similarity 101
Lesson 2 Skills Practice Congruence rite congruence statements comparing the corresponding parts in each set of congruent figures. F 2. F =, FE s J, E s J s ^,_ U ^ ZY, ZS ^_ZT, U s XY, US ^ YT, S ^ TX 3. D s, DC = C, C s C M ^ PL, MK s LQ, K ^ QP 5. 6. M N O z Z s ZD, ZQ s j^,_ 0 s_zl, /Q s DL, QO s SL, ON ^ LD ZM s ZY,^ ^ ^'j^ P^ZX,MN^^YZ,NO^ OP s l/i^x, PM ^ XY = j^l^' Z, 102 Course 3 Chapter 7 Congruence and Similarity
Lesson 2 omework Practice Congruence Triangles C and I are congruent. rite congruence statements comparing the corresponding parts. Then determine which transformation(s) map C onto I. Z ^ Z,_Z ^_Z, ZC s Z/; s, C = I, C ^ /; Sample answer: If you translate C down 5 units and to the right 2 units, It coincides with I. 2U 2. Parallelograms C M P and S I T E are congruent. rite congruence statements comparing the corresponding parts. Then determine which transformation(s) map parallelogram C M P onto parallelogram S I T E. ZC ^ ZS, Z = Zl, Z M =J-'!jJ-P =_ZE; C ^ SI, M s IT, M P s TE, PC s ES; Sample answer: If you rotate parallelogram CMP and then translate it, it coincides with parallelogram SITE. y C s Ps. P E 7 > s / / 0 M X / f J f / f / s / 0 3. Triangles LMN and XYZ are congruent. rite congruence statements comparing the corresponding parts. Then determine which transformation(s) map LMN onto XYZ. ZL s ZX, Z M ^Zy, Z N ^ ZZ; LM s XY, M N ^ YZ, NL s ZX; Sample answer: If you rotate L M N 180 about the origin, it coincides with XYZ. J / L / M y N 0 z X X J / f Y Course 3 Chapter 7 Congruence and Similarity 101
Lesson 3 eteach Similarity and Transformations Two figures are similar if tfie second can be obtained from the first by a sequence of transformations and dilations. ecall that a dilation changes the size of a figure by a scale factor, but does not change the shape of the figure. Example 1 Determine if the two figures are similar by using transformations. U U Since the orientation of the figures is the same, one of the transformations is a translation. rite ratios comparing the lengths of the sides. C _ ± 1 = 1 l ^ = ± n r l M. - l 1 S 8 2' S T 4 2' TU 8 2' U 4 2 S T Since the ratios are equal, CD is the dilated image of STU. So, the two triangles are similar because a translation and a dilation maps CD onto STU. Example 2 Determine if the two figures are similar by using transformations. Since the orientation of the figures is the same, one of the transformations. ^3 C ^2 1 S T ST 4 2 e The ratios are not equal. So, the two triangles are not similar since a dilation did not occur. Exercise Determine if the two figures are similar by using transformations. E The orientation of the figures is the same, so one of the transformations is a translation. Since the ratios of the sides are the same, a dilation is the second transformation and the figures are similar. F V Y X Course 3 Chapter 7 Congruence and Similarity 105
Lesson 3 Skills Practice Similarity and Transformations Determine if the two figures are similar by using transformations. Explain your reasoning. 2. S N 0 0 T Q p no; Sample answer: The ratios of the side lengths are not equal for ail the sides: "^= while 4^= 6 X F 10 no; Sample answer: The ratios of the side lengths are not equal for all the sides; 5 while ^ O 1' O P 10" P 0 4. S L s \ 0 1 J M yes; Sample answer: rotation and a dilation with a scale factor of 2 maps /J onto QOP. no; Sample answer: The ratios of the side lengths are not equal for all the sides; -^ = 1 while-^-^ 106 Course 3 Chapter 7 Congruence and Similarity
Lesson 3 omework Practice Similarity and Transformations Determine if the two figures are similar by using transformations. Explain your reasoning. 2. S \ v. E N k / -- \ / C F u X T V 3. Yes; a dilation of 2 maps C onto EF. \ J \ \ \ s \ s / y K \ L Yes; a rotation of 90 maps STU onto VXT. 4. M P N 1 / / f / / / O Q T S s k s. S No; the ratio of side lengths are not equal; = 8 while-5?- = l. No; the ratio of the length of the parallelograms is not equal to the heights of the parallelograms; NO height of taller 6 = while QT 3 height of shorter 5' 5. MULS Jenna is creating a mural for her bedroom wall. She would like to copy a picture that is 2 inches by 2.5 inches. She uses a copy machine to enlarge it by a factor of 4. Then she projects it on her wall at a factor of 12. h a t are the dimensions of the mural? re the pictures similar? 96 In. by 120 in.; yes 6. IOLOY Mr. Fletcher is looking at a 0.5 millimeter section of plant under a microscope. The plant section appears enlarged by a scale factor of 10 when looking through the microscope. e uses the camera on the microscope to photograph what is seen through the lenses at a scale factor of 20. h a t is the length of the section of plant in the photograph? 100 mm Course 3 Chapter 7 Congruence and Similarity 105