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Proving Triangles are Congruent SSS, SAS; ASA; AAS. CCSS: G.CO7. Standards for Mathematical Practices. 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others.  

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Proving triangles are congruent sss sas asa aas

Proving Triangles are CongruentSSS, SAS; ASA; AAS

CCSS: G.CO7


Standards for mathematical practices
Standards for Mathematical Practices

  • 1. Make sense of problems and persevere in solving them.

  • 2. Reason abstractly and quantitatively.

  • 3. Construct viable arguments and critique the reasoning of others.  

  • 4. Model with mathematics.

  • 5. Use appropriate tools strategically.

  • 6. Attend to precision.

  • 7. Look for and make use of structure.

  • 8. Look for and express regularity in repeated reasoning.


Ccss g co 7
CCSS:G.CO 7

  • USE the definition of congruence in terms of rigid motions to SHOW that two triangles ARE congruent if and only if corresponding pairs of sides and corresponding pairs of angles ARE congruent.


Essential question
ESSENTIAL QUESTION

  • How do we show that triangles are congruent?

  • How do we use triangle congruence to plane and write proves ,and prove that constructions are valid?


Objectives
Objectives:

  • Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem

  • Use congruence postulates and theorems in real-life problems.



SSS AND SASCONGRUENCE POSTULATES

then

If

1.ABDE

4.AD

2.BCEF

5. BE

ABCDEF

3.ACDF

6.CF

If all six pairs of corresponding parts (sides and angles) are

congruent, then the triangles are congruent.

Sides are

congruent

Angles are

congruent

Triangles are

congruent

and


SSS AND SASCONGRUENCE POSTULATES

S

S

S

Side MNQR

then MNPQRS

Side NPRS

Side PMSQ

POSTULATE

POSTULATE 19Side -Side -Side (SSS) Congruence Postulate

If three sides of one triangle are congruent to three sidesof a second triangle, then the two triangles are congruent.

If


SSS AND SASCONGRUENCE POSTULATES

The SSS Congruence Postulate is a shortcut for provingtwo triangles are congruent without using all six pairsof corresponding parts.


SSS AND SASCONGRUENCE POSTULATES

POSTULATE

Side PQWX

A

S

S

then PQSWXY

Angle QX

Side QSXY

POSTULATE 20Side-Angle-Side (SAS) Congruence Postulate

If two sides and the included angle of one triangle are

congruent to two sides and the included angle of a

second triangle, then the two triangles are congruent.

If


Congruent Triangles in a Coordinate Plane

AC FH

ABFG

Use the SSS Congruence Postulate to show that ABCFGH.

SOLUTION

AC = 3 and FH= 3

AB = 5 and FG= 5


Congruent Triangles in a Coordinate Plane

d = (x2 – x1 )2+ (y2 – y1 )2

d = (x2 – x1 )2+ (y2 – y1 )2

BC = (–4 – (–7))2+ (5– 0)2

GH = (6 – 1)2+ (5– 2)2

= 32+ 52

= 52+ 32

= 34

= 34

Use the distance formula to find lengths BC and GH.


Congruent Triangles in a Coordinate Plane

BCGH

BC = 34 and GH= 34

All three pairs of corresponding sides are congruent,

ABCFGH by the SSS Congruence Postulate.


SAS  postulate

SSS  postulate


T

C

S

G

The vertex of the included angle is the point in common.

SSS  postulate

SAS  postulate


SSS  postulate

Not enough info


SSS  postulate

SAS  postulate


Not Enough Info

SAS  postulate


SSS  postulate

Not Enough Info


SAS  postulate

SAS  postulate


Congruent Triangles in a Coordinate Plane

MN DE

PMFE

Use the SSS Congruence Postulate to show that NMPDEF.

SOLUTION

MN = 4 and DE= 4

PM = 5 and FE= 5


Congruent Triangles in a Coordinate Plane

d = (x2 – x1 )2+ (y2 – y1 )2

d = (x2 – x1 )2+ (y2 – y1 )2

PN = (–1 – (– 5))2+ (6– 1)2

FD = (2 – 6)2+ (6– 1)2

= 42+ 52

= (-4)2+ 52

= 41

= 41

Use the distance formula to find lengths PN and FD.


Congruent Triangles in a Coordinate Plane

PNFD

PN = 41 and FD= 41

All three pairs of corresponding sides are congruent,

NMPDEF by the SSS Congruence Postulate.



Postulate 21 angle side angle asa congruence postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.

Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate


Theorem 4 5 angle angle side aas congruence theorem

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem


Theorem 4 5 angle angle side aas congruence theorem1

Given: congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.A  D, C  F, BC  EF

Prove: ∆ABC  ∆DEF

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem


Theorem 4 5 angle angle side aas congruence theorem2

You are given that two angles of congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF.

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem


Ex 1 developing proof

Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Ex. 1 Developing Proof


Ex 1 developing proof1

A. In addition to the angles and segments that are marked, state the postulate or theorem you would use. Explain your reasoning.EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

Ex. 1 Developing Proof


Ex 1 developing proof2

Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Ex. 1 Developing Proof


Ex 1 developing proof3

B. In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

Ex. 1 Developing Proof


Ex 1 developing proof4

Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

UZ ║WX AND UW

║WX.

Ex. 1 Developing Proof

1

2

3

4


Ex 1 developing proof5

The two pairs of parallel sides can be used to show state the postulate or theorem you would use. Explain your reasoning.1  3 and 2  4. Because the included side WZ is congruent to itself, ∆WUZ  ∆ZXW by the ASA Congruence Postulate.

Ex. 1 Developing Proof

1

2

3

4


Ex 2 proving triangles are congruent

Given: AD state the postulate or theorem you would use. Explain your reasoning.║EC, BD  BC

Prove: ∆ABD  ∆EBC

Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC

. Use the fact that AD ║EC to identify a pair of congruent angles.

Ex. 2 Proving Triangles are Congruent


Proof

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

1.

Proof:


Proof1

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

1. Given

Proof:


Proof2

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Proof:


Proof3

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Proof:


Proof4

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

Proof:


Proof5

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

ASA Congruence Theorem

Proof:


Note: state the postulate or theorem you would use. Explain your reasoning.

  • You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D  C and A  E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.


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