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Section 4.3 & 4.4: Proving s are Congruent

Section 4.3 & 4.4: Proving s are Congruent. Goals. Identify  figures and corresponding parts Prove that 2  are . Anchors. Identify and/or use properties of congruent and similar polygons Identify and/or use properties of triangles. M. Q. N. R. P. S.

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Section 4.3 & 4.4: Proving s are Congruent

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  1. Section 4.3 & 4.4: Proving s are Congruent Goals • Identify  figures and corresponding parts • Prove that 2  are  Anchors • Identify and/or use properties of congruent and similar polygons • Identify and/or use properties of triangles

  2. M Q N R P S Side-Side-Side (SSS)  Postulate • If 3 sides of 1  are  to 3 sides of a 2nd , then the 2 ’s are . If Side MN  QR Side NP  RS and Side PM  SQ Then MNP QRS Then we can say: M Q, N  R , and P  S

  3. Statements Reasons Given: W is the midpoint of QS PQ  TS and PW  TWProve: PQW  TSW • W is the mdpt of QS, • PQ  TS and PW  TW • Given 2) QW  SW 2) Def. of midpoint 3) PQW  TSW 3) SSS

  4. Statements Reasons Given: D is the midpoint of ACABC is isosceles ABC is the vertex angleProve: ABD  CBD • D is the mdpt of AC, • ABC is isosceles • Given 2) AD  DC 2) Def. of midpoint 3) AB  BC 3) Property of Isosceles  4) BD  BD 4) Reflexive 5) ABD  CBD 5) SSS

  5. Q X ) P W S Y ) Side-Angle-Side (SAS)  Postulate • If 2 sides and the included  of 1  are  to 2 sides and the included  of a 2nd , then the 2 s are . If Side PQ  WX Angle Q  X Side QS  XY Then PQS WXY Then we can say: PS  WY, P  W , and S  Y

  6. Statements Reasons Given: QRS is isosceles RT bisects QRS QRS is the vertex angle Prove: QRT  SRT ) • QRS is isosceles • RT bisects QRS • Given 2) QRT  SRT 2)  bisector 3) QR  RS 3) Property of Isosceles  4) RT  RT 4) Reflexive 5) QRT  SRT 5) SAS

  7. Statements Reasons Given: BD and AE bisect each otherProve: ABC  EDC ) ) • BD and AE bisect • each other • Given 2) BC  CD, AC  CE 2) Segment bisectors 3) BCA  ECD 3) Vertical angles 4) ABC  EDC 4) SAS

  8. ) Q M ) R N S P ) ) Angle-Side-Angle (ASA)  Postulate • If 2 ’s and the included side of 1  are  to 2 ’s and the included side of a 2nd, then the 2  are  If Angle  N   R Side MN  QR Angle  M   Q Then MNP QRS Then we can say: MP  QS, NP  RS , and P  S

  9. ) ) Statements Reasons Given: B  N RW bisects BNProve: BRO  NWO ) ) • B  N • RW bisects BN • Given 2) BOR  WON 2) Vertical Angles 3) BO  ON 3) Segment bisector 4) BRO  NWO 4) ASA

  10. ) 1 3 4 2 Statements Reasons Given: 1  2 CD bisects BCEProve: BCD  ECD ) ) • 1  2 • CD bisects BCE • Given 2) 3  4 • Supplements of congruent s are congruent 3) BCD  ECD 3) Angle bisector 4) BCE is isosceles 4) Property of isosceles  5) BC  CE 5) Property of isosceles  6) BCD  ECD 6) ASA

  11. X Q ( ( W P Y S ( ( Angle-Angle-Side (AAS)  Theorem • If 2 ’s and a non-included sideof 1  are  to 2  ‘s and a non-included side of a 2nd , then the 2 ’s are . If Angle P  W Angle S  Y Side QP  WX Then PQS WXY Then we can say: QS  XY, PS  WY , and Q  X

  12. ) ) Statements Reasons Given: AD ║ EC , B is the mdpt of CDProve: ABD  EBC ) ) 1) AD ║ EC , B is the mdpt of CD • Given 2) A  E 2) Alternate Interior s 3) ABD  CBE 3) Vertical Angles 4) BD  BC 4) Midpoint 5) ABD  EBC 5) AAS

  13. ) ) Statements Reasons Given: AD ║ EC , B is the mdpt of CDProve: ABD  EBC ) ) 1) AD ║ EC , B is the mdpt of CD • Given 2) A  E, D  C 2) Alternate Interior s 3) BD  BC 3) Midpoint 4) ABD  EBC 4) AAS

  14. 40 40 50 50 Why Angle-Angle-Angle (AAA)Doesn’t Work The angles are , but the sides are proportional.

  15. E ( A D F ( C B Why Side-Side-Angle (SSA)Doesn’t Work Two different triangles can be formed if you use two sides and a non-included angle.

  16. Theorem 4.8: Hypotenuse-Leg (HL)  Theorem • If the hypotenuse and a leg of a right  are  to a hypotenuse and a leg of a 2nd right , then the 2 ’s are  D A If BC  EF and AC  DF, then ABC  DEF Special case of SSA B C E F Then we can say: AB  DE, A  D , and C  F

  17. Statements Reasons Given: RS  QT QRT is isosceles QRT is the vertex angleProve: QRS  TRS 1) RS  QT, QRT is isosceles • Given 2) QSR  90, TSR  90 2) Definition of perpendicular 3) QSR  TSR 3) Substitution 4) QR  RT 4) Property of isosceles  5) RS  RS 5) Reflexive 6) QRS  TRS 6) HL

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