Cholkar MCHS MATH II ___/___/___ Name____________________________ U3L1INV2 What combinations of side or angle measures are sufficient to determine that two triangles are similar? HW # Complete Handout & CYU pg. 172 [CYU, 2, 7, 8] Do Now ACB ~ AED . If AC = 60, CE = 40 and AB = 75, what is BD ? A C E B D INVESTIGATION: SUFFICIENT CONDITIONS FOR SIMILARITY OF TRIANGLES (pg. 168) My role for this investigation _________________________ 1. a. Calculate b. Are your reasoning. uniquely determined? That is, is there exactly one value possible for each? Explain __________________________________________________________________________________________ __________________________________________________________________________________________ c. In general, if you know the values of b, c, and m<A, can values of a, m<B, and m<C always be found? __________________________________________________________________________________________ Could any values of a, m<B, or m<C have two or more values when b, c, and m<A are given? Explain. __________________________________________________________________________________________ d. Summarize your work in Parts a-c in an if-then statement that begins as follows: In a triangle, if the lengths of two sides and the measure of the angle included between those sides are known, then _____________________________________________________________________________________. 2. Next, examine shown below. In both cases, you have information given about two sides and an included angle. In , k is a constant. Test if this information is sufficient to conclude that using Parts a-d as a guide. a. ________________________________________ ________________________________________ b. Use the information given for to write and then solve an equation to find length x. Based on your work in Problem 1, how is x related to a? _________________________________________________________________________________________ c. __________________________________________________________________________________________ d. __________________________________________________________________________________________ 3. a. If so, explain how. If not, explain why not. __________________________________________________________________________________________ __________________________________________________________________________________________ b. Complete the following statement: SIDE-ANGLE-SIDE (SAS) SIMILARITY THEOREM: If an angle of one triangle has the same measure as an angle of a second triangle, and if the lengths of the corresponding side including these angles are multiplied by the same scale factor k, then _________________________________. 4. Suppose you know that in as in the diagram below. the lengths of corresponding sides are related by a scale factor k a. What additional relationship would you need to know in order to conclude that possible strategy you could use to establish the relationship? What is a _________________________________________________________________________________________ b. Write a deductive argument proving the relationship you stated in Part a. What can you conclude? _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ _________________________________________________________________________________________ SIDE-SIDE-SIDE (SSS) SIMILARITY THEOREM: If the lengths of the three sides of one triangle are multiplied by the same scale factor k to obtain the lengths of the three sides of another triangle, then _______________________________. 5. ANGLE-ANGLE (AA) SIMILARITY THEOREM: If the measures of two angles of one triangle are the same as the measures of two angles of another triangle, then ________________________________________. Lesson Summary In this investigation, you established three sets of conditions, each of which is sufficient to prove that two triangles are similar. Math Toolkit Vocabulary: Angle-Angle Similarity Postulate, Side-Side-Side (SSS) Similarity Theorem, Side-Angle-Side (SAS) Similarity Theorem Write a proof for the problem below. Given: JN || LM Prove: JKN MKL Cholkar MCHS MATH II ___/___/___ Name____________________________ HW # CYU pg. 172 1. Use the diagram to the right to complete the statement. a. b. c. d. Determine whether the triangles are similar. If they are, write a similarity statement. 2. 5. 3. 4. 6. The shortest side of a triangle similar to RST is 12 units long. Find the other side lengths of the triangle. REVIEW: 7. Find the values of x and y that make ABC DEF .