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similar_triangles.notebook March 25, 2013 Similar Triangles 1) The student will learn proportional relationships 2) The student will be able to set up proportional relationships between the sides of triangles 3) The student will be able to use and describe AA Similarity Postulate and SSS or SAS Similarity Theorem Sue Blakely SMART Exemplary Educator El Paso, TX Lesson objectives Teachers' notes 1 similar_triangles.notebook March 25, 2013 Subject: Geometry Topic: Similar Triangles Grade(s): 10 Prior knowledge: proportional relationships, fractions, triangles, crossproduct multiplication, congruent angles Lesson notes: Page 8: Let students pull apart triangles and line them up to better see the similarity. Use math tools to compare the congruent angle measurements. (If you don't have math tools, press link and copy of angles will come up.) Page 11: Students fill the appropriate lines of the black triangles with corresponding colors and label the measures. Students drag the letters to the appropriate places on the triangles and angles (cloned letters) Lesson objectives Teachers' notes 2 similar_triangles.notebook March 25, 2013 Ratios Means & Extremes extreme = mean extreme a mean b = mean extreme c mean d extreme 3 similar_triangles.notebook March 25, 2013 Ratios Means & Extremes 15 20 = 30 40 Name the means: Name the extremes: 4 similar_triangles.notebook March 25, 2013 Cross Products Property In a proportion, the product of the extremes equals the product of the means. c a If = where b ≠ 0 and d ≠ 0, then ad = bc d b 2 3 = 4 6 3 x 4 = 2 x 6 = 12 5 similar_triangles.notebook March 25, 2013 Geometric Mean The geometric mean of two positive numbers a and b is the positive number x that satisfies x a 2 b So, x = ab and x = √ab x = . 5 15 Given = , find the geometric mean. 9 3 x = √ab x = √5x9 x = √45 x = 3√5 6 similar_triangles.notebook March 25, 2013 Geometric Mean What is the geometric mean: 2 and 18 4 and 25 2 and 25 6 and 20 7 similar_triangles.notebook March 25, 2013 What makes these two sets of polygons have similar ratios: Explore...... link to angle measures 8 similar_triangles.notebook March 25, 2013 Similar polygons have equal corresponding angle measures 71° 71° 64° 45.1° 64° 45.1° 75.2° 122° 75.2° 122° 59.9° 249.7° 33.2° 59.9° 249.7° 33.2° 9 similar_triangles.notebook March 25, 2013 ΔRST ΔXYZ T R S T X Y Z 20 25 R 18 15 Y 30 S X 12 Z Similar triangles have congruent corresponding angles. If the corresponding angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are similar. 10 similar_triangles.notebook March 25, 2013 ΔZTX ΔRYS Z T X R Y S Arrange the color lines to the appropriate similarity and give the angle congruence. 15 12 = 20 16 R Z Y X ST 11 similar_triangles.notebook March 25, 2013 ΔRST ΔXYZ T R S T X Y Z 20 25 R 18 15 Y 30 S X 12 Z All similar triangles have the same ratio of corresponding sides. (SCALE FACTOR) The Scale Factor is a ratio of one corresponding side of one triangle to the other corresponding side of the second triangle. This is written as 3:5 or 5:3 depending how the triangles are compared. 15 12 18 25 20 30 12 similar_triangles.notebook March 25, 2013 ΔRST ΔXYZ Set up a proportion and solve for x. Give the scale factor T x 25 30 = 15 Y R x 18 30 X S 12 Z 12 Set up corresponding proportion 18 Solve for your variable Scale factor 15:25 reduced is 3:5 13 similar_triangles.notebook March 25, 2013 ΔABC ΔDEF Set up a proportion and solve for x and y. Give the scale factor. F B 12 C x 24 10 8 A D y E 14 similar_triangles.notebook March 25, 2013 15 similar_triangles.notebook March 25, 2013 AngleAngle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are A similar. B C E F D ΔABC ΔDEF 16 similar_triangles.notebook March 25, 2013 AngleAngle (AA) Similarity Postulate A Show ΔABE ΔACD E 52o B (Hint: Redraw the triangles separately) D 52o C 17 similar_triangles.notebook March 25, 2013 SideSideSide (SSS) Similarity Theorem If the corresponding side lengths of two triangles are proportional, then the two triangles are similar. B C A AB BC CA = = DE EF FD ΔABC ΔDEF 8 12 16 E 9 D 6 F 12 4 4 4 = = 3 3 3 18 similar_triangles.notebook March 25, 2013 SideSideSide (SSS) Similarity Theorem Using the SSS Similarity Theorem, explain which set of triangles are similar. Write a similarity statement and give the scale factor. Y 5 10 X Z 20 F 25 D 8 16 M 2 4 L 4 N 10 E 19 similar_triangles.notebook March 25, 2013 SideAngleSide (SAS) Similarity Theorem B If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these 16 angles are proportional, then the triangles are similar. 30o AB AC = DE DF ΔABC ΔDEF A E C 12 12 30o D 9 A D F 4 4 = 3 3 20 similar_triangles.notebook March 25, 2013 SideAngleSide (SAS) Similarity Theorem Explain using the SAS Similarity Theorem why these two triangles are similar. D 20 30 15 B C 5 10 E 10 A ΔABC ΔDEC 21 similar_triangles.notebook March 25, 2013 return to previous page 22