Triangle Congruence and Similarity Proofs

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Triangle Congruence and Similarity Proofs

• Amanda and Terra!

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Introduction
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Introduction!

• Hello, and welcome to the PowerPoint of math
  greatness created by Amanda Rodriguez and Terra
  Berardi!
• Throughout this PowerPoint you will observe many
  geometric proofs of many different shapes and
  sizes, a slide dedicated to the basic components
  of a proof for better understanding, as well as a
  mini tour through the uses of sine, cosine, and
  tangent. For your enjoyment weve also creatively
  named our proofs.
• If there is any trouble understanding a symbol,
  please refer to our glossary which can be reached
  by going to the main menu and selecting the
  glossary button.
• Thank you and we hope you enjoy this PowerPoint!

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Main Menu

• Proof components
• P1 P?Q
• P2 P
• ? Q
• Direct Proof
• P?Q - Positive Statement
• Q ?P - Converse Statement
• Indirect Proof
• Not P ? Not Q - Inverse Statement
• Not Q ? Not P -Contra Positive Statement
• Euclids Axioms
• AB, BC ?? AC
• AB AB
• A-B A-B
• Euclids Postulates

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• Glossary
• Dfn. Definition
• CPCTC- Corresponding parts of congruent triangles
  are congruent
• Alt. Alternate
• Int. Interior
• ? - angle
• ?? - parallel
• ? - perpendicular
• ? - triangle (also known as delta or change)
• ? - congruent
• ? - degree
• ? - therefore
• ? - finished
• - theta (or angle)

Main Menu
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Main page
Introduction

• Proofs
• Proof components
• Fun with Trigonometry!
• Glossary
• Similarities

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Proof Title Triangles- Perpendicular Bisector

• Sketch

Statement
Reason

• ?BAM ? ? CAM
• AB ? AC
• ?B ? ?C
• ? BAM ? ? CAM
• BM ? CM
• ?AMB ? ?AMC
• ?BMC ? 180?
• ?AMB ?AMC ?BMC
• ? AMB 90? ? AMC
• ? AM is ? bis. of BC

• Def. ? bis.
• Given
• ISO. ?
• ASA
• CPCTC
• CPCTC
• Def. Straight ?
• Addition (? ?)
• ? ? Share 180?

Given ? ABC AB ? ACAM is the ? bis. Of ? A
Prove AM is ? of ? A
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Proof Title Bowtie- Jimmy!

• Sketch

Statement
Reason

• ? DCE and ? ACB 90 ?
• AC ? CD
• BC ? CE
• ? ABC ? ?DEC
• ? ABC ? ? DEC
• ? ABC and ? DEC are alt. int. ?s
• ? AB ?? ED

• Dfn. Of ? bis.
• Dfn. Of ? bis.
• Dfn. Of ? bis.
• SAS
• CPCTC
• Dfn. Of Alt. Int. ?s
• Dfn. Of Alt. Int. ?s

Given ? ABC and ? DEC AD and BE are ? bis. Of
each other
Prove AB ?? ED
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Proof Title Rectangle- diagonals of a rectangle
(2-26-07)

• Sketch

Statement
Reason

• AB ? CD
• ?DBA ? ? BDC
• BD ? DB
• ? BDC ? DBA
• ? CB ? AD

• Dfn. Retangle
• Dfn. Rectangle
• Reflexive
• CPCTC

Given ABC
Prove AD ? CB
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Proof Title Triangle Tasha

• Sketch

Statement
Reason

• BM ? CM
• ? BMA 90 ? CMA
• AB ? AC
• ? B ? ? C
• ? AMB ? ? AMC
• ? BAM ? ? CAM
• ? AM is ? bis. of ?BAC

• Dfn. of ? bis.
• given
• iso. ?
• SAS
• CPCTC

Given ?ABC AB ? AC AM ? of BC
Prove AM is ? bisector of ? BAC
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Proof Title Parallelogram- Opposite Angles of a
Parallelogram

• Sketch

Statement
Reason

• ? ABC ? DCB
• CB ? BC
• ? ACB ? DBC
• ? ABC ? ? DCB
• ?A ? ?D
• ?ACB ?BCD ?BDC ?CBA
• ?ACD ? DBA

• Alt. Int. ?s
• Reflexive
• Alt. Int. ?s
• ASA
• CPCTC
• Euclids 2nd Postulate

Given ABCD
Prove ?A ? ?D and ?ACD ? DBA
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Proof Title Parallelogram- Diagonals of a
Parallelogram

• Sketch

Statement
Reason

• ? ABC ? DCB
• CB ? BC
• ? ACB ? DBC
• ? ABC ? ? DCB
• CD ? AB
• ? CED ? BAD
• ? CED ? ? BEA
• CE ? BE
• AE ? ED

• Alt. Int. ?s
• Reflexive
• Alt. Int. ?s
• ASA
• CPCTC
• Alt. Int. ?s
• ASA
• CPCTC
• CPCTC

Given ABCD
Prove CD ? AB and AE ? ED
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Proof Title Triangle Sir Mumsy-Pants

• Sketch

Statement
Reason

• AM ? BC
• AB ? AC
• ? ABM ? ? ACM
• ? BAM ? ? CAM
• ? BAM ? ? CAM
• BM ? CM
• ? MA is ? bis. BC

• Dfn. of Altit.
• given
• iso. ?
• ? sum and sub. Property
• ASA
• CPCTC

Given ? ABC AB ? AC MA is Altit. ? ABC
Prove MA is ? bis. Of BC
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Proof Title parallelogram given

• Sketch

Statement
Reason

• ?ABC ? ?DCB
• CB ? BC
• ?ACB ? ?DCB
• CD ? AB
• ?CDE ? ?BAE
• ? CED ? ? ADE
• CE ? EB
• AE ? ED

• Alt. int. ? s
• reflexive
• Alt. int. ? s
• ASA
• CPCTC
• Alt. int. ? s
• ASA
• CPCTC

Given parallelogram ABCD
Prove AE ? ED and that CE ? EB
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Proof Title bowtie Marty

• Sketch

Statement
Reason

• ? ABE ? ? CDE
• ? ECD ? ? EAB
• ? ?ECD, ?EAB are alt. int. ?s

• SAS
• CPCTC
• Dfn. Alt. int. ?s

Given thing ABCD BE? DE AE ? CE ? BEC 90
Prove ?ECD and ? EAB are alt. int. ?s
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Proof Title parallelogram polliferous

• Sketch

Statement
Reason

• ? ABC ? ? DCB
• CB ? BC
• ? ACB ? ? DBC
• ? ABC ? ? DCB
• CD ? AB
• ? CDE ? ? BAE
• ?CED ? ?BEA
• CE ? EB
• AE ? ED

• Alt. int. ? s
• reflexive
• Alt. int. ? s
• ASA
• CPCTC
• Alt. int. ? s
• ASA
• CPCTC

Given parallelogram ABCD
Prove AE ? ED CE ? EB
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Proof Title Parallelogram- FED

• Sketch

Statement
Reason

• AB ? ? DC
• ?FBD ? ?ECA
• FC ? ? DB
• ?BDF ? ?CAE
• ? ? ACE ? ? DBF

• Dfn.
• Alt. Int. ?s
• Dfn.
• Alt. Int. ?s
• ASA

Given ABCD CE ? BF
Prove ? ACE ? ? DBF
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Proof Title Parallelogram- Parallelogram of DOOM

• Sketch

Statement
Reason

• CB ? BC
• AB ?? DC
• ? ABC ?? ? DCB
• ? CBD ?? ?BCA
• ? CBD ? ?BCA
• CD ? BA
• EF ? FE
• EC ? FB
• EF FB ? FE EC
• ? ? AEB ? ?DFC

• Reflexive
• Dfn. Parallelogram
• Alt. Int. ?s
• Alt. Int. ?s
• ASCPCTCA
• Reflexive
• Given
• Addition
• SAS

Given ABCD CE ? BF
Prove ? AEB ? ?DFC
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Proof Title Vertical Angles-exavieerrre

• Sketch

Statement
Reason

• ? and ? are a linear pair
• ? and ? are supplementary ?s
• Likewise ? ? 180? ? ?
• ? ? ?

• Dfn. Of Linear Pair
• Dfn. Supplementary ?s
• Transitive Property
• Subtraction Property

?
Given To form 4 angles
?
Prove ? ? ?
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Proof Title Triangle- Straight line

• Sketch

Statement
Reason

• ? B ? ? D
• ? C ? ? E
• ? D ? A ? E 180 ?
• ? B ? A ? C 180 ?

• Alt. Int. ?s
• Alt. Int. ?s
• Straight angle/line
• Substitution

Given
? ?
Prove ? B ? A ? C 180?
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Proof Title Intersecting Lines - chubackaaah

• Sketch

Statement
Reason

• Suppose Not
• Then there are 2 intersection points
• ?

• Modus Tolens
• ?

Given
Prove That they dont anywhere else
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Proof Title Intersecting Planes- pasley

• Sketch

Statement
Reason

• Suppose Not
• Then there is at least one point not on the line
  contained by P and P
• Point P makes 3 points which are non-collinear
  on P and P at the same time P ? P
• ? Postulate is true

• Modus Tolens
• Postulate
• Postulate

Given P ? P at Line
Prove That they dont intersect anywhere else
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Proof Title Butterfly- flutter

• Sketch

Statement
Reason

• DE ? EB
• ?D ? ?B
• ?DEA ? ?CEB
• ?AED ? ?CEB
• ?D, ?B are Alt. Int. ?s
• ?AD ? ? BC

• Given
• Given
• Vertical ?s
• ASA
• Dfn. Alt. Int. ?s
• Alt. Int. ?s

Given DE ? EB ?D ? ?B ABCD
Prove AD ? ? BC
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Proof Title Butterfly- schweniquewahs

• Sketch

Statement
Reason

• FE ? FG
• FD ? FH
• ?DFE ? ?HFG
• ?FGH ? ?FED
• ?HGF ? ?DEF
• ?G, ?E are Alt. Int. ?s
• DE ? ? GH

• Dfn. MP
• Dfn. MP
• Vertical ?s
• SAS
• CPCTC
• Dfn. Alt. Int. ?s
• Alt. Int. ?s

Given F is MP of DH and EG
Prove DE ? ? GH
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Proof Title garfunkle

• Sketch

Statement
Reason

• AB ?? DC
• ?FBD ? ?ECA
• AC ? DB
• ?BDF ? ?CAE
• ? ?ACE ? ?DBF

• Dfn. Parallelogram
• Alt. Int. ?s
• Dfn. Parallelogram
• Alt. Int. ?s
• ASA

Given Parallelogram ABCD CE ? BF
Prove ?ACE ? ?DBF
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Proof Title McDugalshmurfpoopsmith

• Sketch

Statement
Reason

• ?AOB ? ?DOE
• AO ? DO
• PO ? OP
• BP ? EP
• ?BPO 90 ? ? EPO
• BO ? EO
• ? ?AOB ? ?DEO

• Vertical ?s
• Given
• Reflexive
• Dfn. MP
• SAS
• CPCTC
• SAS

Given ABDE AO ? DO MPP
Prove ?AOB ? ?DEO
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Proof Title wendurful

• Sketch

Statement
Reason

• ? ABC is a right ?
• CD is altitude to hypotenuse

• Given
• Through a PT. there is one line ? to a given line
• Either leg is geometric mean between hypotenuse
  and adj. segment
• Multiple proportion property
• Either leg is geometric mean between hypotenuse
  and adj. segment
• Multiple proportion property
• Addition property
• Distributive property
• Substitution (if ab then either may be used)

Given ? abc is a right ?
Prove a² b² c²
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Proof Title Similarity- Mickey

• Sketch

Statement
Reason

• X 9
• BDDA
• 9312
• ? BA 12

• CPSTP
• Substitution
• Solve Proportion
• Segment addition
• Substitution
• Transitive prop.

Given ? ABC ?DBE
Prove BA12
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Proof Title Similarity- my-hine

• Sketch

Statement
Reason

• ?AEB ? ? DEC
• ?A ? ?D
• ?A, ?D Alt. Int. ?s
• ? AB ? ? DC

• 15 times 44 20 times 33
• Vertical ?s
• SAS
• CPSTP
• Dfn. Alt. Int. ?s
• Alt. Int. ?s are ?

Given The Image above
Prove AB ? ? DC
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Proof Title periwinkle

• Sketch

Statement
Reason

• ?ACB ? ? QCD
• x4.5
• AC QC
• y12
• BC DC
• ? ? ABC ? QDC

• Vertical Angles
• proportions
• solve proportion
• 3644.5
• proportions
• solve proportion
• 94312
• SAS
• ?

Given AB 3 DQ 4 BC 9 QC 6
Prove ? ABC ? QDC
Continue to Similarities
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SimilaritiesSolve for x!
Do the proportions match? Lets find out!
Step one
Step two
multiply
divide
Step three-solve!
Yes they do!
X36
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Continue to fun with trig
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Fun with Trigonometry!
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• What is Trigonometry?
• According to dictionary.com, trigonometry is
  branch of mathematics that deals with relations
  between sides and angles of triangles

• What is Trigonometry used for?
• Trig is most commonly used for finding angles or
  sides of triangles. The next few slides will give
  a brief demonstration of this.

A33.69? B56.31? I90 ?
Key Terms theta (or angle) ? delta (or
change)
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TANGENTS

• The formula
• To find the hypotenuse of the triangle
• Substitute the variables for the numbers
• tan30?

The hypotenuse of this triangle is 5.774
Then find the value of X x5tan30?
x5.5774 x2.887
x
Once you find the value, use the Pythagorean
theorem to find the hypotenuse. a²b²c²
5²2.887²c²
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COSINES

• The formula
• To find the opposite side of the triangle
• Substitute the variables for the numbers
• Cos30?

The adjacent side of this triangle is 5
Then find the value of X x5.774cos30?
x5.774.866 x5
X
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SINE

• The formula
• To find the opposite side of the triangle
• Substitute the variables for the numbers
• sin30?

The opposite side of this triangle is 2.887
Then find the value of X x5.774tan30?
x5.774.5 x2.887
5.774
x
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Conclusion