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Are They Congruent?

HOMEWORK: WS - Congruent Triangles. Proving Δ ’ s are  using: SSS, SAS, HL, ASA, & AAS. Are They Congruent?. SSS. If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. Methods of Proving Triangles Congruent. SAS.

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Are They Congruent?

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  1. HOMEWORK: WS - Congruent Triangles Proving Δ’s are using: SSS, SAS, HL, ASA, & AAS Are They Congruent?

  2. SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. Methods of Proving Triangles Congruent SAS If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. ASA If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. AAS If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. HL If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.

  3. GIVEN KL NL,KM NM PROVE KLMNLM DIRECT Information Direct information comes in two forms: congruent statements in the ‘GIVEN:’ part of a proof marked in the picture Example: OR

  4. INDIRECT Information Indirect Information appears in the ‘GIVEN:’ part of the proof but is NOT a congruency statement Example: J Given: JO  SH; O is the midpoint of SH Prove:  SOJ  HOJ S O H

  5. INDIRECT Information • Perpendicular lines  right angles  all rt∠s are ≅ • Midpoint of a segment  2 ≅ segments • Parallel lines  AIA • Parallelogram  2 sets of parallel lines  2 pairs of AIA • Segment is an angle bisector  2 ≅ angles • Segments bisect each other  2 sets of ≅ segments • Perpendicular bisector of a segment  2 ≅ segments & • 2 right angles

  6. BUILT-IN Information Built- in information is part of the drawing. Example: Vertical angles  VA Shared side  Reflexive Property Shared angle  Reflexive Property Any Parallelogram 2 pairs parallel lines  2 pairs of AIA

  7. Steps to Write a Proof • Take the 1st Given and MARK it on the picture • WRITE this Given in the PROOF & its reason • If the Given is NOT a ≅ statement, • write the ≅stmt to match the marksContinue until there are no more GIVEN • 4. Do you have 3 ≅ statements? • If not, look for BUILT-IN parts • 5. Do you have ≅ triangles? • If not, write CNBD • If YES, Write the triangle congruency • and reason (SSS, SAS, SAA, ASA, HL)

  8. GIVEN KL NL,KM NM PROVE KLMNLM ≅ ≅ ≅  given  given  reflexive prop ΔKLM≅ΔNLMSSS

  9. BC DA,BC AD BC DA BC AD ACAC ≅ GIVEN ΔABC ≅ΔCDA PROVE given ≅ given ≅ AIA ∠BCA∠DAC ≅ reflexive prop SAS ΔABC ≅ΔCDA

  10. D Given: A  D, C  F,  Prove: ∆ABC  ∆DEF A B F A  D given C E C  Fgiven  given ∆ABC  ∆DEFAAS

  11. Given: bisects IJK, ILJ   JLK Prove: ΔILJ  ΔKLJ bisects IJK Given IJL  IJH Definition of angle bisector ILJ   JLK Given  Reflexive Prop ΔILJ  ΔKLJ ASA

  12. Given: , Prove: ΔTUV  ΔWXV Given Given TVU  WVX Vertical angles  ΔTUV  ΔWXV SAS

  13. Given: , H L Prove: ΔHIJ  ΔLKJ Given H L Given IJH  KJL Vertical angles  ΔHIJ  ΔLKJ ASA

  14. Given: , PRT  STR Prove: ΔPRT  ΔSTR Given PRT  STR Given Reflexive Prop ΔPRT  ΔSTR SAS

  15. Given: is perpendicular bisector of Prove: is perpendicular bisector of given ∠ABM & ∠PBM are rt ∠s def lines ∠ABM ≅ ∠PBM all rt ∠s are ≅ ≅ def bisector ≅ reflexive prop. ΔABM ≅ ΔPBM SAS

  16. Given: O is the midpoint of and Prove: ΔMON ≅ ΔPOQ O is the midpoint of and given ≅def. midpoint ≅def. midpoint ∠MON ≅∠ VA ΔMON ≅Δ SAS

  17. Given: ≅ ; || Prove: ΔABD ≅ ΔCDB ≅ given || given ∠ADB ≅ ∠CBD AIA ≅ reflexive prop. ΔABD ≅ ΔCDB SAS

  18. Given: ; O is the midpoint of Prove:  SOJ  HOJ

  19. Given: HJ  GI, GJ  JIProve: ΔGHJ  ΔIHJ

  20. Given: 1  2; A  E ; C is midpt of AEProve: ΔABC  ΔEDC

  21. Given:  ,  , and  Prove: ΔPQR  ΔPSR  Given PQR = 90° Def.  lines  Given PSR = 90° Def.  lines PQR  PSR all right s are   Given  Reflexive Prop ΔPQR  ΔPSR HL

  22. Checkpoint Decide if enough information is given to prove the triangles are congruent. If so, state the congruence postulate you would use.

  23. Given: LJ bisects IJK, ILJ   JLK Prove: ΔILJ  ΔKLJ

  24. Given: 1  2, A  E and  Prove: ΔABC  ΔEDC 1  2Given A  E Given  Given ΔABC  ΔEDC ASA

  25. Given:  ,  Prove: ΔABD  ΔCBD  Given  Given  Reflexive Prop  ΔABD  ΔCBD SSS

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