Prove It!

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