Prove It!

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Prove It!

  1. 1. Prove it! Geometry Proofs
  2. 2. Prove it! Givens and Conclusions
  3. 3. Givens and conclusions <ul><li>In geometry proofs, you are told certain things. These are the GIVENS. We can assume that these are always true for the problem. </li></ul><ul><li>Your results based on the GIVENS are the CONCLUSION(S) </li></ul><ul><li>Given: AC  CB </li></ul><ul><li>Conclusion: C is the midpoint of AB </li></ul>
  4. 4. Prove it! Triangle congruencies
  5. 5. Triangle congruencies <ul><li>Two shapes are congruent if all their sides and angles are congruent </li></ul><ul><li>For Δ ABC  Δ DEF you need to know that: </li></ul><ul><li>AC  DF </li></ul><ul><li>CB  DE </li></ul><ul><li>AB  FE </li></ul><ul><li> CAB   DFE </li></ul><ul><li> ACB   FDE </li></ul><ul><li> CBA   DEF </li></ul>
  6. 6. Triangle congruencies <ul><li>There are five shortcuts! </li></ul>
  7. 7. Prove it! Triangle congruency short cuts
  8. 8. Triangle congruency short-cuts <ul><li>If you can prove just one of the following short cuts, you have two congruent triangles </li></ul><ul><li>SSS (side-side-side) </li></ul><ul><li>SAS (side-angle-side) </li></ul><ul><li>ASA (angle-side-angle) </li></ul><ul><li>AAS (angle-angle-side) </li></ul><ul><li>HL (hypotenuse-leg) right triangles only! </li></ul>
  9. 9. Triangle congruency short-cuts <ul><li>Given: Δ ABC and Δ DEF, AC  DF, CB  DE, AB  FE </li></ul><ul><li>Conclusion: Δ ABC  Δ DEF because of SSS </li></ul>
  10. 10. Prove it! Writing 2-column Proofs
  11. 11. Writing 2-column Proofs <ul><li>The left column lists the statements you are making </li></ul><ul><li>The right column lists the reasons why you are making the statements </li></ul><ul><li>Your final conclusion should be what you are trying to prove </li></ul>
  12. 12. 2-column Proof example <ul><li>Given: Δ GHI, HJ  GI, GJ  JI </li></ul><ul><li>Prove: Δ GHJ  Δ IHJ </li></ul><ul><li>Statements : Reasons : </li></ul><ul><li>GJ  JI Given </li></ul><ul><li> GJH,  IJH = 90° HJ  GI </li></ul><ul><li> GJH   IJH Both angles = 90° </li></ul><ul><li>HJ  HJ Both triangles share the same side </li></ul><ul><li> Δ GHJ  Δ IHJ SAS </li></ul>
  13. 13. 2-column Proof <ul><li>Given: Δ ABC, Δ EDC,  1   2, </li></ul><ul><li> A   E and AC  EC </li></ul><ul><li>Prove: Δ ABC  Δ EDC </li></ul><ul><li>Statements : Reasons : </li></ul><ul><li> 1   2 Given </li></ul><ul><li> A   E Given </li></ul><ul><li>AC  EC Given </li></ul><ul><li> Δ ABC  Δ EDC ASA </li></ul>
  14. 14. 2-column Proof <ul><li>Given: Δ ABD, Δ CBD, AB  CB, </li></ul><ul><li>and AD  CD </li></ul><ul><li>Prove: Δ ABD  Δ CBD </li></ul><ul><li>Statements : Reasons : </li></ul><ul><li>AB  CB Given </li></ul><ul><li>AD  CD Given </li></ul><ul><li>BD  BD Both triangles share the same side </li></ul><ul><li> Δ ABD  Δ CBD SSS </li></ul>
  15. 15. 2-column Proof <ul><li>Given: LJ bisects  IJK, </li></ul><ul><li> ILJ   JLK </li></ul><ul><li>Prove: Δ ILJ  Δ KLJ </li></ul><ul><li>Statements : Reasons : </li></ul><ul><li> ILJ   JLK Given </li></ul><ul><li> IJL   IJH Definition of bisector </li></ul><ul><li>JL  JL Both triangles share the same side </li></ul><ul><li> Δ ILJ  Δ KLJ ASA </li></ul>
  16. 16. 2-column Proof <ul><li>Given: Δ TUV, Δ WXV, TV  VW, </li></ul><ul><li>UV  VX </li></ul><ul><li>Prove: Δ TUV  Δ WXV </li></ul><ul><li>Statements : Reasons : </li></ul><ul><li>TV  VW Given </li></ul><ul><li>UV  VX Given </li></ul><ul><li> TVU   WVX Definition of vertical angles </li></ul><ul><li> Δ TUV  Δ WXV SAS </li></ul>
  17. 17. 2-column Proof <ul><li>Given: Given: HJ  JL,  H  L </li></ul><ul><li>Prove: Δ HIJ  Δ LKJ </li></ul><ul><li>Statements : Reasons : </li></ul><ul><li>HJ  JL Given </li></ul><ul><li> H  L Given </li></ul><ul><li> IJH   KJL Definition of vertical angles </li></ul><ul><li> Δ HIJ  Δ LKJ ASA </li></ul>
  18. 18. 2-column Proof <ul><li>Given: Quadrilateral PRST with PR  ST, </li></ul><ul><li> PRT   STR </li></ul><ul><li>Prove: Δ PRT  Δ STR </li></ul><ul><li>Statements : Reasons : </li></ul><ul><li>PR  ST Given </li></ul><ul><li> PRT   STR Given </li></ul><ul><li>RT  RT Both triangles share the same side </li></ul><ul><li> Δ PRT  Δ STR SAS </li></ul>
  19. 19. 2-column Proof <ul><li>Given: Quadrilateral PQRS, PQ  QR, </li></ul><ul><li>PS  SR, and QR  SR </li></ul><ul><li>Prove: Δ PQR  Δ PSR </li></ul><ul><li>Statements : Reasons : </li></ul><ul><li>PQ  QR Given </li></ul><ul><li> PQR = 90° PQ  QR </li></ul><ul><li>PR is a hypotenuse Hypotenuse is opposite 90° angle </li></ul><ul><li>PS  SR Given </li></ul><ul><li> PSR = 90° PS  SR </li></ul><ul><li>PR  PR Both triangles share the hypotenuse </li></ul><ul><li>QR  SR Given </li></ul><ul><li>Δ PQR  Δ PSR HL </li></ul>
  20. 20. Prove it! NOT triangle congruency short cuts
  21. 21. NOT triangle congruency short-cuts <ul><li>The following are NOT short cuts: </li></ul><ul><li>AAA (angle-angle-angle) </li></ul><ul><li>Triangles are similar but not necessarily congruent </li></ul>
  22. 22. NOT triangle congruency short-cuts <ul><li>The following are NOT short cuts </li></ul><ul><li>SSA (side-side-angle) </li></ul><ul><li>SAS is a short cut but the angle is in between both sides! </li></ul>
  23. 23. Prove it! CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  24. 24. CPCTC <ul><li>Once you have proved two triangles congruent using one of the short cuts, the rest of the parts of the triangle you haven’t proved directly are also congruent! </li></ul><ul><li>We say: Corresponding Parts of Congruent Triangles are Congruent or CPCTC for short </li></ul>
  25. 25. CPCTC example <ul><li>Given: Δ TUV, Δ WXV, TV  WV, </li></ul><ul><li>TW bisects UX </li></ul><ul><li>Prove: TU  WX </li></ul><ul><li>Statements : Reasons : </li></ul><ul><li>TV  WV Given </li></ul><ul><li>UV  VX Definition of bisector </li></ul><ul><li> TVU   WVX Vertical angles are congruent </li></ul><ul><li>Δ TUV  Δ WXV SAS </li></ul><ul><li> TU  WX CPCTC </li></ul>

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