UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 6: Defining and Applying Similarity Instruction
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1 Prerequisite Skills This lesson requires the use of the following skills: understanding that the sum of the measures of the angles in a triangle is 180 identifying both corresponding and congruent parts of triangles Introduction When a series of similarity transformations are performed on a triangle, the result is a similar triangle. When triangles are similar, the corresponding angles are congruent and the corresponding sides are of the same proportion. It is possible to determine if triangles are similar by measuring and comparing each angle and side, but this can take time. There exists a set of similarity statements, similar to the congruence statements, that let us determine with less information whether triangles are similar. Key oncepts The ngle-ngle () Similarity Statement is one statement that allows us to prove triangles are similar. The Similarity Statement allows that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. XYZ Y Notice that it is not necessary to show that the third pair of angles is congruent because the sum of the angles must equal 180. Similar triangles have corresponding sides that are proportional. The ngle-ngle Similarity Statement can be used to solve various problems, including those that involve indirect measurement, such as using shadows to find the height of tall structures. X Z U1-349
2 ommon Errors/Misconceptions misidentifying congruent parts because of the orientation of the triangles misreading similarity statements as congruent statements incorrectly creating proportions between corresponding sides U1-350
3 Guided Practice Example 1 Explain why the triangles are similar and write a similarity statement. D 1. Identify the given information. ccording to the diagram, D and E. E 2. State your conclusion. DE by the ngle-ngle () Similarity Statement. Example 2 Explain why DEF, and then find the length of DF. 3.4 E D F 1. Show that the triangles are similar. ccording to the diagram, D and F. DEF by the ngle-ngle () Similarity Statement. U1-351
4 2. Find the length of DF. orresponding sides of similar triangles are proportional. reate and solve a proportion to find the length of DF. DE DF x (2.72)(6.9) (3.4)(x) Solve for x x x 5.52 The length of DF is 5.52 units. orresponding sides are proportional. Substitute known values. Let x represent the length of DF. Example 3 Identify the similar triangles. Find x and the measures of the indicated sides. x G J x + 5 H 6 1. Show that the triangles are similar. ccording to the diagram, H and J. HGJ by the ngle-ngle () Similarity Statement. U1-352
5 2. Use the definition of similar triangles to find the value of x. orresponding sides of similar triangles are proportional. reate and solve a proportion to find the value of x. orresponding sides are proportional. DE DF x+ 1 x+ 5 Substitute known values. 3 6 (x + 1)(6) (3)(x + 5) Solve for x. 6x + 6 3x x 9 x 3 3. Find the unknown side lengths. Use the value of x to find the unknown lengths of the triangles. x x The length of is 4 units. The length of is 8 units. U1-353
6 Example 4 Suppose a person 5 feet 10 inches tall casts a shadow that is 3 feet 6 inches long. t the same time of day, a flagpole casts a shadow that is 12 feet long. To the nearest foot, how tall is the flagpole? 1. Identify the known information. The height of a person and the length of the shadow cast create a right angle. The height of the flagpole and the length of the shadow cast create a second right angle. You can use this information to create two triangles. Draw a picture to help understand the information. x 5 ft 10 in 3 ft 6 in 12 ft 2. Determine if the triangles are similar. Two pairs of angles are congruent. ccording to the ngle-ngle () Similarity Statement, the triangles are similar. orresponding sides of similar triangles are proportional. U1-354
7 3. Find the height of the flagpole. reate and solve a proportion to find the height of the flagpole x x 12 Simplify. 5 1 ( )( 12) ( 3 )( x) Solve for x ( )( x) x 20 The flagpole is 20 feet tall. 2 orresponding sides are proportional. Let x represent the height of the flagpole. U1-355
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