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Prove it!
Geometry Proofs
Prove it!
Givens and Conclusions
Givens and conclusions
In geometry proofs, you are told certain things. These are the GIVENS. We can assume that these are always true for the problem.
Your results based on the GIVENS are the CONCLUSION(S)
Given: AC CB Conclusion: C is the midpoint
of AB
C
A
B
Prove it!
Triangle congruencies
Triangle congruencies Two shapes are congruent if
all their sides and angles are congruent
For ΔABCΔDEF you need to know that:
1. ACDF
2. CBDE
3. ABFE
4. CABDFE
5. ACBFDE
6. CBADEF
A B
C
D
EF
Triangle congruencies
There are five shortcuts!
Prove it!
Triangle congruency short cuts
Triangle congruency short-cuts
If you can prove just one of the following short cuts, you have two congruent triangles
1. SSS (side-side-side)
2. SAS (side-angle-side)
3. ASA (angle-side-angle)
4. AAS (angle-angle-side)
5. HL (hypotenuse-leg) right triangles only!
Triangle congruency short-cuts
Given: ΔABC and ΔDEF, ACDF, CBDE, ABFE
Conclusion: ΔABCΔDEF because of SSS
A B
C
D
EF
Prove it!
Writing 2-column Proofs
Writing 2-column Proofs
The left column lists the statements you are making
The right column lists the reasons why you are making the statements
Your final conclusion should be what you are trying to prove
2-column Proof example
Given: ΔGHI, HJ GI, GJ JI
Prove: ΔGHJ ΔIHJ
Statements: Reasons:
1. GJ JI Given
2. GJH, IJH = 90°HJ GI
3. GJH IJH Both angles = 90°
4. HJ HJ Both triangles share the same side
5. ΔGHJ ΔIHJ SAS
JG
H
I
2-column Proof
Given: ΔABC, ΔEDC, 1 2,
A E and AC EC
Prove: ΔABC ΔEDC
Statements: Reasons:
1. 1 2 Given
2. A E Given
3. AC EC Given
4. ΔABC ΔEDC ASA
21
C
D
EA
B
2-column Proof
Given: ΔABD, ΔCBD, AB CB,
and AD CD
Prove: ΔABD ΔCBD
Statements: Reasons:
1. AB CB Given
2. AD CD Given
3. BD BD Both triangles share the same side
4. ΔABD ΔCBD SSS
B
C
A
D
2-column Proof
Given: LJ bisects IJK,
ILJ JLK
Prove: ΔILJ ΔKLJ
Statements: Reasons:
1. ILJ JLK Given
2. IJL IJH Definition of bisector
3. JL JL Both triangles share the same side
4. ΔILJ ΔKLJ ASA
J
K
I
L
2-column Proof
Given: ΔTUV, ΔWXV, TV VW,
UV VX
Prove: ΔTUV ΔWXV
Statements: Reasons:
1. TV VW Given
2. UV VX Given
3. TVU WVX Definition of vertical angles
4. ΔTUV ΔWXV SAS
VT
W
U
X
2-column Proof
Given: Given: HJ JL, H L
Prove: ΔHIJ ΔLKJ
Statements: Reasons:
1. HJ JL Given
2. H L Given
3. IJH KJL Definition of vertical angles
4. ΔHIJ ΔLKJ ASA
L
J
KI
H
2-column Proof
Given: Quadrilateral PRST with PR ST,
PRT STR
Prove: ΔPRT ΔSTR
Statements: Reasons:
1. PR ST Given
2. PRT STR Given
3. RT RT Both triangles share the same side
4. ΔPRT ΔSTR SAS
S
P T
R
2-column ProofGiven: Quadrilateral PQRS, PQ QR,
PS SR, and QR SR Prove: ΔPQR ΔPSRStatements: Reasons:1. PQ QR Given2. PQR = 90° PQ QR 3. PR is a hypotenuse Hypotenuse is opposite 90° angle4. PS SR Given5. PSR = 90°PS SR 6. PR PR Both triangles share the hypotenuse7. QR SR Given8. ΔPQR ΔPSR HL
S
RP
Q
Prove it!
NOT triangle congruency short cuts
NOT triangle congruency short-cuts
The following are NOT short cuts:
AAA (angle-angle-angle)
Triangles are similar but not necessarily congruent
60
60
60
A
BC
60
60
60
D
F E
NOT triangle congruency short-cuts
The following are NOT short cuts
SSA (side-side-angle) SAS is a short cut but
the angle is in between both sides!
5 cm8 cm
34
A
B
C
5 cm8 cm
34
D
E
F
Prove it!
CPCTC (Corresponding Parts of Congruent Triangles are
Congruent)
CPCTC
Once you have proved two triangles congruent using one of the short cuts, the rest of the parts of the triangle you haven’t proved directly are also congruent!
We say: Corresponding Parts of Congruent Triangles are Congruent or CPCTC for short
CPCTC example
Given: ΔTUV, ΔWXV, TV WV,
TW bisects UX
Prove: TU WX
Statements: Reasons:
1. TV WV Given
2. UV VX Definition of bisector
3. TVU WVX Vertical angles are congruent
4. ΔTUV ΔWXV SAS
5. TU WX CPCTC
VT
W
U
X
