Geometry unit 7.3
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This document discusses proving triangles are similar using the AA, SSS, and SAS similarity criteria. It provides examples of using these criteria to prove triangles are similar, find missing side lengths in similar triangles using proportionality, and write proofs involving similar triangles. Applications to engineering problems are also presented. The key methods covered are using corresponding angles or sides that are equal to prove triangles are similar, setting up proportions between corresponding sides in similar triangles to solve for unknowns, and structuring multi-statement proofs involving similar triangles.
Geometry unit 7.3
- 1. UUNNIITT 77..33 PPRROOVVIINNGG TTRRIIAANNGGLLEESS SSIIMMIILLAARR
- 2. Warm Up Solve each proportion. 1. 2. 3. z = ±10 x = 8 4. If ΔQRS ~ ΔXYZ, identify the pairs of congruent angles and write 3 proportions using pairs of corresponding sides. ÐQ @ ÐX; ÐR @ ÐY; ÐS @ ÐZ;
- 3. Objectives Prove certain triangles are similar by using AA, SSS, and SAS. Use triangle similarity to solve problems.
- 4. There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.
- 5. Example 1: Using the AA Similarity Postulate Explain why the triangles are similar and write a similarity statement. Since , ÐB @ ÐE by the Alternate Interior Angles Theorem. Also, ÐA @ ÐD by the Right Angle Congruence Theorem. Therefore ΔABC ~ ΔDEC by AA~.
- 6. Check It Out! Example 1 Explain why the triangles are similar and write a similarity statement. By the Triangle Sum Theorem, mÐC = 47°, so ÐC @ ÐF. ÐB @ ÐE by the Right Angle Congruence Theorem. Therefore, ΔABC ~ ΔDEF by AA ~.
- 9. Example 2A: Verifying Triangle Similarity Verify that the triangles are similar. ΔPQR and ΔSTU Therefore ΔPQR ~ ΔSTU by SSS ~.
- 10. Example 2B: Verifying Triangle Similarity Verify that the triangles are similar. ΔDEF and ΔHJK ÐD @ ÐH by the Definition of Congruent Angles. Therefore ΔDEF ~ ΔHJK by SAS ~.
- 11. Check It Out! Example 2 Verify that ΔTXU ~ ΔVXW. ÐTXU @ ÐVXW by the Vertical Angles Theorem. Therefore ΔTXU ~ ΔVXW by SAS ~.
- 12. Example 3: Finding Lengths in Similar Triangles Explain why ΔABE ~ ΔACD, and then find CD. Step 1 Prove triangles are similar. ÐA @ ÐA by Reflexive Property of @, and ÐB @ ÐC since they are both right angles. Therefore ΔABE ~ ΔACD by AA ~.
- 13. Example 3 Continued Step 2 Find CD. Corr. sides are proportional. Seg. Add. Postulate. Substitute x for CD, 5 for BE, 3 for CB, and 9 for BA. x(9) = 5(3 + 9) Cross Products Prop. 9x = 60 Simplify. Divide both sides by 9.
- 14. Check It Out! Example 3 Explain why ΔRSV ~ ΔRTU and then find RT. Step 1 Prove triangles are similar. It is given that ÐS @ ÐT. ÐR @ ÐR by Reflexive Property of @. Therefore ΔRSV ~ ΔRTU by AA ~.
- 15. Check It Out! Example 3 Continued Step 2 Find RT. Corr. sides are proportional. Substitute RS for 10, 12 for TU, 8 for SV. Cross Products Prop. Simplify. Divide both sides by 8. RT(8) = 10(12) 8RT = 120 RT = 15
- 16. Example 4: Writing Proofs with Similar Triangles Given: 3UT = 5RT and 3VT = 5ST Prove: ΔUVT ~ ΔRST
- 17. Example 4 Continued Statements Reasons 1. 3UT = 5RT 1. Given 2. 2. Divide both sides by 3RT. 3. 3VT = 5ST 3. Given. 4. 4. Divide both sides by3ST. 5. ÐRTS @ ÐVTU 5. Vert. Ðs Thm. 6. ΔUVT ~ ΔRST 6. SAS ~ Steps 2, 4, 5
- 18. Check It Out! Example 4 Given: M is the midpoint of JK. N is the midpoint of KL, and P is the midpoint of JL.
- 19. Check It Out! Example 4 Continued Statements Reasons 1. M is the mdpt. of JK, 1. Given N is the mdpt. of KL, and P is the mdpt. of JL. 2. 2. Δ Midsegs. Thm 3. 3. Div. Prop. of =. 4. ΔJKL ~ ΔNPM 4. SSS ~ Step 3
- 20. Example 5: Engineering Application The photo shows a gable roof. AC || FG. ΔABC ~ ΔFBG. Find BA to the nearest tenth of a foot. BF » 4.6 ft. BA = BF + FA » 6.3 + 17 » 23.3 ft Therefore, BA = 23.3 ft.
- 21. Check It Out! Example 5 What if…? If AB = 4x, AC = 5x, and BF = 4, find FG. Corr. sides are proportional. Substitute given quantities. Cross Prod. Prop. Simplify. 4x(FG) = 4(5x) FG = 5
- 22. You learned in Chapter 2 that the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. These properties also hold true for similarity of triangles.
- 23. Lesson Quiz 1. Explain why the triangles are similar and write a similarity statement. 2. Explain why the triangles are similar, then find BE and CD.
- 24. Lesson Quiz 1. By the Isosc. Δ Thm., ÐA @ ÐC, so by the def. of @, mÐC = mÐA. Thus mÐC = 70° by subst. By the Δ Sum Thm., mÐB = 40°. Apply the Isosc. Δ Thm. and the Δ Sum Thm. to ΔPQR. mÐR = mÐP = 70°. So by the def. of @, ÐA @ ÐP, and ÐC @ ÐR. Therefore ΔABC ~ ΔPQR by AA ~. 2. ÐA @ ÐA by the Reflex. Prop. of @. Since BE || CD, ÐABE @ ÐACD by the Corr. Ðs Post. Therefore ΔABE ~ ΔACD by AA ~. BE = 4 and CD = 10.
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