ARE THEY CONGRUENT?
HOMEWORK:
WS - Congruent Triangles
Proving Δ’s are using: SSS, SAS, HL, ASA, & AAS
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
DIRECT InformationDirect information comes in two forms:
congruent statements in the ‘GIVEN:’ part of a proofmarked in the picture
Example:
GIVEN KL NL, KM NM
PROVE KLM NLMOR
INDIRECT Information
Indirect Information appears in the ‘GIVEN:’ part of the proof but is NOT a congruency statement
Example:Given: JO SH; O is the midpoint of SH Prove: SOJ HOJ
J
HOS
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
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 Proof1. Take the 1st Given and MARK it on the picture2. WRITE this Given in the PROOF & its reason3. If the Given is NOT a ≅ statement,
write the ≅ stmt to match the marks Continue until there are no more GIVEN4. Do you have 3 ≅ statements?
If not, look for BUILT-IN parts5. Do you have ≅ triangles?
If not, write CNBDIf YES, Write the triangle congruency and reason (SSS, SAS, SAA, ASA, HL)
GIVEN KL NL, KM NM
PROVE KLM NLM
≅ ≅≅
ΔKLM ≅ ΔNLM SSS
given
given
reflexive prop
GIVEN
PROVE
BC DA, BC AD
BC DA
BC AD
∠BCA ∠DAC
AC AC
given
AIA
reflexive prop
ΔABC ≅ ΔCDA
given
SAS
ΔABC ≅ ΔCDA
≅
≅
≅
≅
Given: A D, C F, Prove: ∆ABC ∆DEF
A B
C
D
E
FA Dgiven
C F given
given
∆ABC ∆DEF AAS
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
J
K
I
L
Given: ,
Prove: ΔTUV ΔWXV
Given
Given
TVU WVX Vertical angles
ΔTUV ΔWXV SAS
VT
W
U
X
Given: , H L
Prove: ΔHIJ ΔLKJ
Given
H L Given
IJH KJL Vertical angles
ΔHIJ ΔLKJ ASA
L
J
KI
H
Given: , PRT STR
Prove: ΔPRT ΔSTR
Given
PRT STR Given
Reflexive Prop
ΔPRT ΔSTR SAS
S
P T
R
Given: is perpendicular bisector of
Prove:
is perpendicular bisector of given
∠ABM & PBM are rt s∠ ∠ def lines
≅ def bisector
∠ABM PBM ≅ ∠ all rt s are ∠ ≅
≅ reflexive prop.
ΔABM ΔPBM ≅ SAS
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
Given: ≅ ; ||
Prove: ΔABD ≅ ΔCDB
≅ given
|| given
∠ADB CBD≅ ∠ AIA
≅ reflexive prop.
ΔABD ΔCDB≅ SAS
Given: ; O is the midpoint of Prove: SOJ HOJ
J
S H0
Given: HJ GI, GJ JIProve: ΔGHJ ΔIHJ
JG
H
I
Given: 1 2; A E ; C is midpt of AEProve: ΔABC ΔEDC
21
C
D
EA
B
Given: , , and Prove: ΔPQR ΔPSR
Given PQR = 90° Def. lines Given PSR = 90° Def. linesPQR PSR all right s are Given Reflexive Prop
ΔPQR ΔPSR HL
S
RP
Q
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
J
K
I
L
Given: 1 2, A E and Prove: ΔABC ΔEDC
1 2 Given
A E Given
Given
ΔABC ΔEDC ASA
21
C
D
EA
B
Given: , Prove: ΔABD ΔCBD
Given
Given
Reflexive Prop
ΔABD ΔCBD SSS
B
C
A
D