4 4, 4-5 congruent triangles
TRANSCRIPT
Jim Smith JCHSJim Smith JCHS
Sections 4-3, 4-5Sections 4-3, 4-5
When we talk about congruent triangles,When we talk about congruent triangles,we mean everything about them is congruent. we mean everything about them is congruent. All 3 pairs of corresponding angles are equal….All 3 pairs of corresponding angles are equal….
And all 3 pairs of corresponding sides are equalAnd all 3 pairs of corresponding sides are equal
For us to prove that 2 people are For us to prove that 2 people are identical twins, we don’t need to show identical twins, we don’t need to show that all “2000” body parts are equal. We that all “2000” body parts are equal. We can take a short cut and show 3 or 4 can take a short cut and show 3 or 4 things are equal such as their face, age things are equal such as their face, age and height. If these are the same I think and height. If these are the same I think we can agree they are twins. The same we can agree they are twins. The same is true for triangles. We don’t need to is true for triangles. We don’t need to prove all 6 corresponding parts are prove all 6 corresponding parts are congruent. We have 5 short congruent. We have 5 short cuts or methods.cuts or methods.
SSSSSSIf we can show all 3 pairs of corr.If we can show all 3 pairs of corr.
sides are congruent, the triangles sides are congruent, the triangles have to be congruent.have to be congruent.
SASSASShow 2 pairs of sides and the Show 2 pairs of sides and the
included angles are congruent and included angles are congruent and the triangles have to be congruent.the triangles have to be congruent.
IncludedIncludedangleangle
Non-includedNon-includedanglesangles
This is called a common side.This is called a common side.It is a side for both triangles.It is a side for both triangles.
We’ll use the reflexive property.We’ll use the reflexive property.
Which method can be used toWhich method can be used toprove the triangles are congruentprove the triangles are congruent
Common sideCommon side
SSSSSS
Parallel linesParallel linesalt int anglesalt int angles
Common sideCommon side
Vertical anglesVertical angles
SASSAS
SASSAS
ASA, AAS and HL ASA, AAS and HL
ASA – 2 anglesASA – 2 anglesand the included sideand the included side
AA
SS
AAAAS – 2 angles andAAS – 2 angles andThe non-included sideThe non-included side AAAA
SS
HLHL ( hypotenuse leg ) is used ( hypotenuse leg ) is usedonly with right triangles, BUT, only with right triangles, BUT,
not all right triangles. not all right triangles.
HLHL ASAASA
When Starting A Proof, Make TheWhen Starting A Proof, Make TheMarks On The Diagram IndicatingMarks On The Diagram Indicating
The Congruent Parts. Use The GivenThe Congruent Parts. Use The GivenInfo, Properties, Definitions, Etc. Info, Properties, Definitions, Etc.
We’ll Call Any Given Info That Does We’ll Call Any Given Info That Does Not Specifically State Congruency Not Specifically State Congruency
Or Equality A Or Equality A PREREQUISITEPREREQUISITE
SOME REASONS WE’LL BE USINGSOME REASONS WE’LL BE USING
• DEF OF MIDPOINTDEF OF MIDPOINT
• DEF OF A BISECTORDEF OF A BISECTOR
• VERT ANGLES ARE CONGRUENTVERT ANGLES ARE CONGRUENT
• DEF OF PERPENDICULAR BISECTORDEF OF PERPENDICULAR BISECTOR
• REFLEXIVE PROPERTY (COMMON SIDE)REFLEXIVE PROPERTY (COMMON SIDE)
• PARALLEL LINES ….. ALT INT ANGLESPARALLEL LINES ….. ALT INT ANGLES
AA
BB
CC
DDEE
11 22
Given: AB = BDGiven: AB = BD EB = BCEB = BC
Prove: Prove: ∆ABE ∆ABE ˜ ˜ ∆DBC∆DBC==
SASSASOur OutlineOur OutlineP P rerequisitesrerequisitesS S idesidesA A nglesnglesS S idesidesTriangles ˜Triangles ˜ ==
AA CC
DD
Given: AB = BDGiven: AB = BD EB = BCEB = BC
Prove: Prove: ∆ABE ∆ABE ˜ ˜ ∆DBC∆DBC==BB
EE
11 22
SASSAS
nonenoneAB = BD Given AB = BD Given 1 = 2 Vertical angles1 = 2 Vertical anglesEB = BC GivenEB = BC Given∆∆ABE ˜ ∆DBC SASABE ˜ ∆DBC SAS==
STATEMENTS REASONSSTATEMENTS REASONS
PPSSAASS∆’∆’ss
AA BB
CC
11 22
Given: CX bisects ACBGiven: CX bisects ACB A A ˜ B˜ BProve: Prove: ∆ACX∆ACX ˜ ∆BCX˜ ∆BCX
XX
====
AASAAS
PPAAAASS∆’∆’ss
CX bisects ACB GivenCX bisects ACB Given 1 = 2 1 = 2 Def Def of angle biscof angle bisc A = B GivenA = B Given CX = CX Reflexive PropCX = CX Reflexive Prop∆∆ACX ˜ ∆BCX AASACX ˜ ∆BCX AAS==
Can you prove these trianglesCan you prove these trianglesare congruent?are congruent?
AA BB
DD CC
XX
Given: AB ll DCGiven: AB ll DCX is the midpoint of ACX is the midpoint of ACProve: AXB Prove: AXB ˜ CXD˜ CXD==
AA BB
DD CC
XX
Given: AB ll DCGiven: AB ll DCX is the midpoint of ACX is the midpoint of ACProve: AXB Prove: AXB ˜ CXD˜ CXD==
ASAASA