Unit 5 Similar Triangles
Lesson: Similar Triangles Just as congruence introduced us to new notation, similarity will have its own set of notation. If ΔCAT is congruent to ΔMEW, we write CAT MEW. If two polygons are similar we use the symbol that sits above the equal sign in the congruent symbol: ~. For example, the statement CORN~ PEAS says that quadrilateral CORN is similar to quadrilateral PEAS. Just as in statements of congruence, the order of the letters tells you which segments and which angles in the two polygons correspond. Corresponding angles are congruent: Corresponding segments are proportional: The ratio of the lengths of any two segments in one polygon is equal to the ratio of the corresponding two segments in the similar polygon. For example, CO PE NR SA or OR EA CO PE As we know, Similar Figures have the same shape, but not necessarily the same size. All corresponding angles are equal and corresponding sides are proportional. Proportionality is based on a scale factor which we will see later in transformation (dilations). Guided Practice Directions: Use the similarity statement to solve for x. 1. ΔUTV~ΔLKM 2. ΔUVW~ΔMKL
3. ΔABC~ΔLKJ 4. ΔSRT~ΔERF 5. GFE GUV 6. BCD BQR 7. CTS CED 8. JCD JLK
Skills Practice: Similar Triangles Directions: Given the similarity statement, solve for the missing variable. 1. ΔCDE~ΔUTS 2. ΔSRT~ΔBCA 3. ΔRSQ~ΔDEC 4. ΔRSQ~ΔGFH
5. ΔLKM~ΔQPR 6. ΔGHF~ΔVWU 7. ΔJKL~ΔJCD 8. ΔQRS~ΔNML
9. ΔMNR~ΔQPR 10. ΔTSR~ΔCAB 11. UBS UST Answers to Skill Practice: 1) 8 2) 12 3) 9 4) 5 5) 4 6) 12 7) 6 8) 12 9) 10 10) 7 11) 9
Additional Practice- Similar Triangles 1. Triangle ABC is similar to triangle PQR. Write a proportion that can be used to find n. Then solve for n. 2. Given ABC DEF, circle the following true proportions/statements. AB AC DE EF AC DF BC EF AB BC AC EF BC DF CB EF C F A E C D A D 3. Triangle MNO and PQR are similar a. List corresponding angles b. List the ratios of corresponding sides.
Warm Up 1. Triangles ABC and DEF are similar. a. List the ratios of corresponding sides. b. Find the lengths of the missing sides. 2. Triangles GHI and JKL are similar. a. List the ratios of corresponding sides. b. Find the lengths of the missing sides. c. Find the measures of the missing angles.
Complete the following graphic organizer: Proving Triangles are Similar Mathematically Term/ Postulate Definition/ Explanation Diagram Side Side - Side SSS~ Statement: Side-Angle-Side SAS~ Statement:
Angle- Angle AA~ Statement: Guided Practice: Directions: Show that the following triangles are similar, by showing either angles congruent or sides proportional. Next state the similarity statement. 1. 2. Show corresponding parts: Show corresponding parts: Circle: AA, SSS, SAS Circle: AA, SSS, SAS Statement: ABC Statement: GHJ
3. Show corresponding parts: Circle: AA, SSS, SAS Statement: FHG
4. Determine if the two triangles are similar. If so, state the similarity statement and state how you know they are similar (SSS~, SAS~, AA~). Justify your conclusions. a. b. 5. Multiple Choice: In the triangle shown, GH DF What is the length of GE? A. 2.0 B. 4.5 C. 7.5 D. 8.0 12
Homework- Similar Triangles For questions 1 4, write a similarity statement. Then find the measures of the missing sides. 13
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11. Use the figure below to answer the following questions. a. ABE by b. Solve for x. c. Solve for b given that a = 13. 15
12. Which of the following is true about the triangles below? A. Similar but not congruent B. Congruent but not similar C. Both similar and congruent D. Neither similar nor congruent 13. Which of the following is true about the triangles below? A. Similar but not congruent B. Congruent but not similar C. Both similar and congruent D. Neither similar nor congruent 14. SAT Prep: A summer camp counselor wants to find a length, x, in feet across a lake as represented in the sketch below. The lengths represented by AB, EB, BD, and CD on the sketch were determined to be 1800 feet, 1400 feet, 700 feet, and 800 feet respectively. Segments AC and DE intersect at B, and AEB and CDB have the same measure. What is the value of x? 16
Identify which property will prove these triangles are similar (AA similarity, SAS similarity, SSS similarity) 17
Warm- Up State if the triangles in each pair are similar. If so, state how you know they are similar and complete the similarity statement. 1. 2. 3. SAT Prep: In the figure below, AE CD and segment AD intersects segment CE at B. What is the length of segment CE? 18
Complete the given two-column proofs. Lesson: Similar Triangle Proofs NO MO 1. Given QO PO Prove: MNO ~ PQO Statements Reasons 2. Given: MQ OP Prove: MNQ ~ PON Statements Reasons 19
3. Given: AE BD Prove: ACE ~ BCD 4. Given DAB and DCA are right triangles Prove: DAB ~ DCA Statements Reasons 20
Skills Practice: Proving Similar Triangles 1. Given: m T m M Prove: NTI ~ NMA 2. Given: M P, O Q Prove: OMN QPR GH GI 3. Given:, G J KJ JL Prove: GHI JKL 21