Maintaining Mathematical Proficiency

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1 Name ate hapter 8 Maintaining Mathematical Proficiency Tell whether the ratios form a proportion , , , , , , Find the scale factor of the dilation L 9 L M N M N 220 Geometry opyright ig Ideas Learning, LL

2 Name ate 8.1 Similar Polygons For use with xploration 8.1 ssential Question How are similar polygons related? 1 XPLORTION: omparing Triangles after a ilation Go to igideasmath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software to draw any. ilate to form a similar using any scale factor k and any center of dilation. a. ompare the corresponding angles of and. b. Find the ratios of the lengths of the sides of to the lengths of the corresponding sides of. What do you observe? c. Repeat parts (a) and (b) for several other triangles, scale factors, and centers of dilation. o you obtain similar results? opyright ig Ideas Learning, LL Geometry 221

3 Name ate 8.1 Similar Polygons (continued) 2 XPLORTION: omparing Triangles after a ilation Go to igideasmath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software to draw any. ilate to form a similar using any scale factor k and any center of dilation. a. ompare the perimeters of and. What do you observe? b. ompare the areas of and. What do you observe? c. Repeat parts (a) and (b) for several other triangles, scale factors, and centers of dilation. o you obtain similar results? ommunicate Your nswer. How are similar polygons related? 4. RST is dilated by a scale factor of to form RST. 1 square inch. What is the area of RST? The area of RST is 222 Geometry opyright ig Ideas Learning, LL

4 Name ate 8.1 Notetaking with Vocabulary For use after Lesson 8.1 In your own words, write the meaning of each vocabulary term. similar figures similarity transformation corresponding parts ore oncepts orresponding Parts of Similar Polygons In the diagram below, is similar to F. You can write is similar to F as ~ F. similarity transformation preserves angle measure. So, corresponding angles are congruent. similarity transformation also enlarges or reduces side lengths by a scale factor k. So, corresponding side lengths are proportional. a b c similarity transformation F ka kb kc orresponding angles Ratios of corresponding side lengths,, F F F = = = k Notes: opyright ig Ideas Learning, LL Geometry 22

5 Name ate 8.1 Notetaking with Vocabulary (continued) orresponding Lengths in Similar Polygons If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons. Notes: Theorems Theorem 8.1 Perimeters of Similar Polygons If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. If KLMN ~ PQRS, then K N L M P S Q R PQ + QR + RS + SP PQ QR RS SP = = = =. KL + LM + MN + NK KL LM MN NK Notes: Theorem 8.2 reas of Similar Polygons If two polygons are similar, then the ratio of their areas is equal to the squares of the ratios of their corresponding side lengths. If KLMN ~ PQRS, then K N L M P S Q R rea of PQRS PQ QR RS SP = = = =. rea of KLMN KL LM MN NK Notes: 224 Geometry opyright ig Ideas Learning, LL

6 Name ate 8.1 Notetaking with Vocabulary (continued) xtra Practice In xercises 1 and 2, the polygons are similar. Find the value of x x 12 x 2 20 In xercises 8, ~ KLMNP.. Find the scale factor from to KLMNP. K Find the scale factor from KLMNP to. 5. Find the values of x, y, and z. 2 z L x x N M y P 6. Find the perimeter of each polygon. 7. Find the ratio of the perimeters of to KLMNP. 8. Find the ratio of the areas of to KLMNP. opyright ig Ideas Learning, LL Geometry 225

7 Name ate 8.2 Proving Triangle Similarity by For use with xploration 8.2 ssential Question What can you conclude about two triangles when you know that two pairs of corresponding angles are congruent? 1 XPLORTION: omparing Triangles Go to igideasmath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software. a. onstruct and F so that m = m = 106, m = m = 1, and F is not congruent to F b. Find the third angle measure and the side lengths of each triangle. Record your results in column 1 of the table below m, m m, m m m F F F 226 Geometry opyright ig Ideas Learning, LL

8 Name ate 8.2 Proving Triangle Similarity by (continued) 1 XPLORTION: omparing Triangles (continued) c. re the two triangles similar? xplain. d. Repeat parts (a) (c) to complete columns 2 and of the table for the given angle measures. e. omplete each remaining column of the table using your own choice of two pairs of equal corresponding angle measures. an you construct two triangles in this way that are not similar? f. Make a conjecture about any two triangles with two pairs of congruent corresponding angles. ommunicate Your nswer 2. What can you conclude about two triangles when you know that two pairs of corresponding angles are congruent?. Find RS in the figure at the right. M N R L T 4 S opyright ig Ideas Learning, LL Geometry 227

9 Name ate 8.2 Notetaking with Vocabulary For use after Lesson 8.2 In your own words, write the meaning of each vocabulary term. similar figures similarity transformation Theorems Theorem 8. ngle-ngle () Similarity Theorem If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. If and, then ~ F. F Notes: 228 Geometry opyright ig Ideas Learning, LL

10 Name ate 8.2 Notetaking with Vocabulary (continued) xtra Practice In xercises 1 and 2, determine whether the triangles are similar. If they are, write a similarity statement. xplain your reasoning F F In xercises and 4, show that the two triangles are similar.. 4. P Q S R opyright ig Ideas Learning, LL Geometry 229

11 Name ate 8.2 Notetaking with Vocabulary (continued) In xercises 5 1, use the diagram to complete the statement. 6 6 G F 5. m G = 6. m G = 7. m G = 8. G = 9. = 10. F = 11. = 12. GF = 1. G ~ 14. Using the diagram for xercises 5 1, write similarity statements for each triangle similar to FG. 15. etermine if it is possible for HJK and PQR to be similar. xplain your reasoning. m H = 100, m K = 46, m P = 44, and m Q = Geometry opyright ig Ideas Learning, LL

12 Name ate 8. Proving Triangle Similarity by SSS and SS For use with xploration 8. ssential Question What are two ways to use corresponding sides of two triangles to determine that the triangles are similar? 1 XPLORTION: eciding Whether Triangles re Similar Go to igideasmath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software. a. onstruct and F with the side lengths given in column 1 of the table below F F m m m m m m F b. omplete column 1 in the table above. c. re the triangles similar? xplain your reasoning. d. Repeat parts (a) (c) for columns 2 6 in the table. e. How are the corresponding side lengths related in each pair of triangles that are similar? Is this true for each pair of triangles that are not similar? opyright ig Ideas Learning, LL Geometry 21

13 Name ate 8. Proving Triangle Similarity by SSS and SS (continued) 1 XPLORTION: eciding Whether Triangles re Similar (continued) f. Make a conjecture about the similarity of two triangles based on their corresponding side lengths. g. Use your conjecture to write another set of side lengths of two similar triangles. Use the side lengths to complete column 7 of the table. 2 XPLORTION: eciding Whether Triangles re Similar Go to igideasmath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software. onstruct any. a. Find,, and m. hoose any positive rational number k and construct F so that = k, F = k, and m = m. b. Is F similar to? xplain your reasoning. c. Repeat parts (a) and (b) several times by changing and k. escribe your results. ommunicate Your nswer. What are two ways to use corresponding sides of two triangles to determine that the triangles are similar? 22 Geometry opyright ig Ideas Learning, LL

14 Name ate 8. Notetaking with Vocabulary For use after Lesson 8. In your own words, write the meaning of each vocabulary term. similar figures corresponding parts slope parallel lines perpendicular lines Theorems Theorem 8.4 Side-Side-Side (SSS) Similarity Theorem If the corresponding side lengths of two triangles are proportional, then the triangles are similar. If = =, then ~ RST. RS ST TR S R T Notes: opyright ig Ideas Learning, LL Geometry 2

15 Name ate 8. Notetaking with Vocabulary (continued) Theorem 8.5 Side-ngle-Side (SS) Similarity Theorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. ZX XY If X M and =, then PM MN XYZ ~ MNP. X Z M Y P N Notes: xtra Practice In xercises 1 and 2, determine whether RST is similar to S 2. 7 T R T 2 S R 24 Geometry opyright ig Ideas Learning, LL

16 Name ate 8. Notetaking with Vocabulary (continued). Find the value of x that makes RST ~ HGK. R S G x + 7 H x T K 4. Verify that RST ~ XYZ. Find the scale factor of RST to XYZ. RST : RS = 12, ST = 15, TR = 24 XYZ : XY = 28, YZ = 5, ZX = 56 In xercises 5 and 6, use. 5. The shortest side of a triangle similar to is 15 units long. Find the other side lengths of the triangle The longest side of a triangle similar to is 6 units long. Find the other side lengths of the triangle. opyright ig Ideas Learning, LL Geometry 25

17 Name ate 8.4 Proportionality Theorems For use with xploration 8.4 ssential Question What proportionality relationships exist in a triangle intersected by an angle bisector or by a line parallel to one of the sides? 1 XPLORTION: iscovering a Proportionality Relationship Go to igideasmath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software to draw any. a. onstruct parallel to with endpoints on and, respectively. b. ompare the ratios of to and to. c. Move to other locations parallel to with endpoints on and, and repeat part (b). d. hange and repeat parts (a) (c) several times. Write a conjecture that summarizes your results. 26 Geometry opyright ig Ideas Learning, LL

18 Name ate 8.4 Proportionality Theorems (continued) 2 XPLORTION: iscovering a Proportionality Relationship Go to igideasmath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software to draw any. a. isect and plot point at the intersection of the angle bisector and. b. ompare the ratios of to and to. c. hange and repeat parts (a) and (b) several times. Write a conjecture that summarizes your results. ommunicate Your nswer. What proportionality relationships exist in a triangle intersected by an angle bisector or by a line parallel to one of the sides? 4. Use the figure at the right to write a proportion. opyright ig Ideas Learning, LL Geometry 27

19 Name ate 8.4 Notetaking with Vocabulary For use after Lesson 8.4 In your own words, write the meaning of each vocabulary term. corresponding angles ratio proportion Theorems Theorem 8.6 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Q S T U R RT RU Notes: If TU QS, then =. TQ US Theorem 8.7 onverse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. RT RU Notes: If =, then TU QS. TQ US Q S T U R 28 Geometry opyright ig Ideas Learning, LL

20 Name ate 8.4 Notetaking with Vocabulary (continued) Theorem 8.8 Three Parallel Lines Theorem If three parallel lines intersect two transversals, then they divide the transversals proportionally. Notes: r s U W Y V X Z t m UW WY = VX XZ Theorem 8.9 Triangle ngle isector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. Notes: = xtra Practice In xercises 1 and 2, find the length of opyright ig Ideas Learning, LL Geometry 29

21 Name ate 8.4 Notetaking with Vocabulary (continued) In xercises and 4, determine whether XY.. Z 4. Z X X Y Y In xercises 5 7, use the diagram to complete the proportion. U X V Y W Z 5. UV XY UW = 6. XY YZ = 7. VW ZY = WU WV In xercises 8 and 9, find the value of the variable b 9 x Geometry opyright ig Ideas Learning, LL

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