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

2. 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

3. 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

4. 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

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 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 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)

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

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

10. ∆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

11. 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

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

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

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

15. 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

16. 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

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

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

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

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

21. 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

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. 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

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