Triangle Congruence and Similarity Proofs - PowerPoint PPT Presentation

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Triangle Congruence and Similarity Proofs

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Throughout this PowerPoint you will observe many geometric ... Proof Title: Triangle Sir Mumsy-Pants. Sketch. AM BC. AB AC. ABM ACM. BAM CAM. BAM CAM. BM CM ... – PowerPoint PPT presentation

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Title: Triangle Congruence and Similarity Proofs


1
Triangle Congruence and Similarity Proofs
  • Amanda and Terra!

Main Menu
Introduction
2
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!

Main Menu
3
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

4
  • 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
5
Main page
Introduction
  • Proofs
  • Proof components
  • Fun with Trigonometry!
  • Glossary
  • Similarities

6
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
7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
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
15
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
16
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
17
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
18
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 ? ? ?
19
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?
20
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
21
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
22
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
23
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
24
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
25
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
26
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
27
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
28
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
29
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
Main Menu
30
SimilaritiesSolve for x!
Do the proportions match? Lets find out!
Step one
Step two
multiply
divide
Step three-solve!
Yes they do!
X36
Main Menu
Continue to fun with trig
31
Fun with Trigonometry!
Main Menu
  • 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)
32
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
33
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
34
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
Main Menu
Conclusion
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