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

Methods of Proving Triangles Congruent SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are 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. Methods of Proving Triangles Congruent

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

Example: 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

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

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

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 marks Continue 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)

ΔKLM ≅ ΔNLM SSS GIVEN KL NL, KM NM ≅ ≅ ≅ PROVE KLM NLM 𝐾𝐿  𝑁𝐿 given 𝐿𝑀  𝐿𝑀 reflexive prop ΔKLM ≅ ΔNLM SSS

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

∆ABC  ∆DEF AAS Given: A  D, C  F, 𝐵𝐶  𝐸𝐹 Prove: ∆ABC  ∆DEF D A  D given C E C  F given 𝐵𝐶  𝐸𝐹 given ∆ABC  ∆DEF AAS

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

 ΔTUV  ΔWXV SAS Given: 𝑇𝑊 ≅ 𝑉𝑊 , 𝑈𝑉 ≅ 𝑉𝑋 Prove: ΔTUV  ΔWXV TVU  WVX Vertical angles  ΔTUV  ΔWXV SAS

 ΔHIJ  ΔLKJ ASA Given: 𝐻𝐽 ≅ 𝐽𝐿 , H L Prove: ΔHIJ  ΔLKJ H L Given IJH  KJL Vertical angles  ΔHIJ  ΔLKJ ASA

ΔPRT  ΔSTR SAS Given: 𝑃𝑅 ≅ 𝑆𝑇 , PRT  STR Prove: ΔPRT  ΔSTR PRT  STR Given 𝑅𝑇 ≅ 𝑅𝑇 Reflexive Prop ΔPRT  ΔSTR SAS

ΔABM ≅ ΔPBM SAS Given: 𝑀𝐵 is perpendicular bisector of 𝐴𝑃 Prove: ∆𝐴𝐵𝑀≅ ∆𝑃𝐵𝑀 𝑀𝐵 is perpendicular bisector of 𝐴𝑃 given ∠ABM & ∠PBM are rt ∠s def  lines ∠ABM ≅ ∠PBM all rt ∠s are ≅ 𝐴𝐵 ≅ 𝐵𝑃 def  bisector 𝐵𝑀 ≅ 𝐵𝑀 reflexive prop. ΔABM ≅ ΔPBM SAS

ΔMON ≅ ΔQOP SAS Given: O is the midpoint of 𝑀𝑄 and 𝑁𝑃 Prove: ΔMON ≅ ΔPOQ O is the midpoint of 𝑀𝑄 and 𝑁𝑃 given 𝑀𝑂 ≅ 𝑂𝑄 def. midpoint 𝑁𝑂 ≅ 𝑂𝑃 def. midpoint ∠MON ≅ ∠QOP VA ΔMON ≅ ΔQOP SAS

ΔABD ≅ ΔCDB SAS Given: 𝐴𝐷 ≅ 𝐶𝐷 ; 𝐴𝐷 || 𝐶𝐷 Prove: ΔABD ≅ ΔCDB ∠ADB ≅ ∠CBD AIA 𝐵𝐷 ≅ 𝐵𝐷 reflexive prop. ΔABD ≅ ΔCDB SAS

Given: 𝐽𝑂  𝑆𝐻 ; O is the midpoint of 𝑆𝐻 Prove:  SOJ  HOJ

Given: HJ  GI, GJ  JI Prove: ΔGHJ  ΔIHJ

Given: 1  2; A  E ; C is midpt of AE Prove: ΔABC  ΔEDC

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

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

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

ΔABC  ΔEDC ASA Given: 1  2, A  E and 𝐴𝐶  𝐸𝐶 Prove: ΔABC  ΔEDC 1  2 Given A  E Given 𝐴𝐶  𝐸𝐶 Given ΔABC  ΔEDC ASA

Given: 𝐴𝐵  𝐶𝐵 , 𝐴𝐷  𝐶𝐷 Prove: ΔABD  ΔCBD 𝐴𝐵  𝐶𝐵 Given 𝐴𝐷  𝐶𝐷 Given 𝐵𝐷  𝐵𝐷 Reflexive Prop  ΔABD  ΔCBD SSS