UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 7: Proving Similarity Instruction
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1 Prerequisite Skills This lesson requires the use of the following skills: creating ratios solving proportions identifying both corresponding and congruent parts of triangles Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The ngle-ngle () Similarity Statement is one of them. The Side-ngle-Side (SS) and Side-Side-Side (SSS) similarity statements are two more ways to show that triangles are similar. In this lesson, we will prove that triangles are similar using the similarity statements. Key oncepts The Side-ngle-Side (SS) Similarity Statement asserts that if the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. Similarity statements identify corresponding parts just like congruence statements do. m n mx nx = (x) = (x) U1-377
2 The Side-Side-Side (SSS) Similarity Statement asserts that if the measures of the corresponding sides of two triangles are proportional, then the triangles are similar. m n mx p px nx = (x) = (x) = (x) It is important to note that while both similarity and congruence statements include an SSS and an SS statement, the statements do not mean the same thing. Similar triangles have corresponding sides that are proportional, whereas congruent triangles have corresponding sides that are of the same length. Like with the ngle-ngle Similarity Statement, both the Side-ngle-Side and the Side-Side- Side similarity statements can be used to solve various problems. The ability to prove that triangles are similar is essential to solving many problems. proof is a set of justified statements organized to form a convincing argument that a given statement is true. efinitions, algebraic properties, and previously proven statements can be used to prove a given statement. There are several types of proofs, such as paragraph proofs, two-column proofs, and flow diagrams. U1-378
3 very good proof includes the following: a statement of what is to be proven a list of the given information if possible, a diagram including the given information step-by-step statements that support your reasoning ommon rrors/misconceptions misidentifying congruent parts because of the orientation of the triangles misreading similarity statements as congruency statements incorrectly creating proportions between corresponding sides U1-379
4 Guided Practice xample 1 Prove Lengths for each side of both triangles are given. = 4 = = 8 = = 9 = ompare the side lengths of both triangles. Pair the lengths of the sides of with the corresponding lengths of the sides of to determine if there is a common ratio Notice the common ratio, 2 ; the side lengths are proportional State your conclusion. Similar triangles must have side lengths that are proportional. by the Side-Side-Side (SSS) Similarity Statement. U1-380
5 xample 2 etermine whether the triangles are similar. xplain your reasoning ccording to the diagram,. Given the side lengths, both and are included angles. 2. ompare the given side lengths of both triangles. If the triangles are similar, then the corresponding sides are proportional The side lengths are proportional. 3. State your conclusion. The measures of two sides of are proportional to the measures of two corresponding sides of, and the included angles are congruent. by the Side-ngle-Side (SS) Similarity Statement. U1-381
6 xample 3 etermine whether the triangles are similar. xplain your reasoning The measures of each side of both triangles are given. 2. ompare the side lengths of both triangles. Pair the lengths of the sides of with the corresponding lengths of the sides of to determine if there is a common ratio Notice there is not a common ratio; therefore, the side lengths are not proportional. 3. State your conclusion. Similar triangles must have side lengths that are proportional. is not similar to. U1-382
7 xample 4 Identify the similar triangles and then find the value of x x and are both right angles; therefore,. and are corresponding sides. and are corresponding sides. by the Side-ngle-Side (SS) Similarity Statement. 2. etermine the scale factor of the triangle sides The scale factor is ind the length of x = + x+ 2.5 x Solve the proportion = for x. x (4.5)(2) = (1)(x + 2.5) ind the cross products. 9 = x Simplify. x =.5 Solve for x. The length of x is.5 units. U1-383
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3.2 etermining Similarity O evelop strategies to determine if two shapes are similar. YOU WI N a ruler a protractor RN OUT the Math ia-wen and Sam are sorting isosceles triangles to solve a puzzle in a