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Section 4.3 & 4.4: Proving s are CongruentPowerPoint Presentation

Section 4.3 & 4.4: Proving s are Congruent

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### Section 4.3 & 4.4: Proving s are Congruent

Goals

- Identify figures and corresponding parts
- Prove that 2 are

Anchors

- Identify and/or use properties of congruent and similar polygons
- Identify and/or use properties of triangles

Q

N

R

P

S

Side-Side-Side (SSS) Postulate- If 3 sides of 1 are to 3 sides of a 2nd , then the 2 ’s are .

If Side MN QR

Side NP RS and

Side PM SQ

Then MNP QRS

Then we can say: M Q, N R , and P S

Reasons

Given: W is the midpoint of QS PQ TS and PW TWProve: PQW TSW- W is the mdpt of QS,
- PQ TS and PW TW

- Given

2) QW SW

2) Def. of midpoint

3) PQW TSW

3) SSS

Reasons

Given: D is the midpoint of ACABC is isosceles ABC is the vertex angleProve: ABD CBD- D is the mdpt of AC,
- ABC is isosceles

- Given

2) AD DC

2) Def. of midpoint

3) AB BC

3) Property of Isosceles

4) BD BD

4) Reflexive

5) ABD CBD

5) SSS

X

)

P

W

S

Y

)

Side-Angle-Side (SAS) Postulate- If 2 sides and the included of 1 are to 2 sides and the included of a 2nd , then the 2 s are .

If Side PQ WX

Angle Q X

Side QS XY

Then PQS WXY

Then we can say: PS WY, P W , and S Y

Reasons

Given: QRS is isosceles RT bisects QRS QRS is the vertex angle Prove: QRT SRT)

- QRS is isosceles
- RT bisects QRS

- Given

2) QRT SRT

2) bisector

3) QR RS

3) Property of Isosceles

4) RT RT

4) Reflexive

5) QRT SRT

5) SAS

Reasons

Given: BD and AE bisect each otherProve: ABC EDC)

)

- BD and AE bisect
- each other

- Given

2) BC CD, AC CE

2) Segment bisectors

3) BCA ECD

3) Vertical angles

4) ABC EDC

4) SAS

Q

M

)

R

N

S

P

)

)

Angle-Side-Angle (ASA) Postulate- If 2 ’s and the included side of 1 are to 2 ’s and the included side of a 2nd, then the 2 are

If Angle N R

Side MN QR

Angle M Q

Then MNP QRS

Then we can say: MP QS, NP RS , and P S

)

Statements

Reasons

Given: B N RW bisects BNProve: BRO NWO)

)

- B N
- RW bisects BN

- Given

2) BOR WON

2) Vertical Angles

3) BO ON

3) Segment bisector

4) BRO NWO

4) ASA

1

3

4

2

Statements

Reasons

Given: 1 2 CD bisects BCEProve: BCD ECD)

)

- 1 2
- CD bisects BCE

- Given

2) 3 4

- Supplements of congruent s are congruent

3) BCD ECD

3) Angle bisector

4) BCE is isosceles

4) Property of isosceles

5) BC CE

5) Property of isosceles

6) BCD ECD

6) ASA

Q

(

(

W

P

Y

S

(

(

Angle-Angle-Side (AAS) Theorem- If 2 ’s and a non-included sideof 1 are to 2 ‘s and a non-included side of a 2nd , then the 2 ’s are .

If Angle P W

Angle S Y

Side QP WX

Then PQS WXY

Then we can say: QS XY, PS WY , and Q X

)

Statements

Reasons

Given: AD ║ EC , B is the mdpt of CDProve: ABD EBC)

)

1) AD ║ EC , B is the mdpt of CD

- Given

2) A E

2) Alternate Interior s

3) ABD CBE

3) Vertical Angles

4) BD BC

4) Midpoint

5) ABD EBC

5) AAS

)

Statements

Reasons

Given: AD ║ EC , B is the mdpt of CDProve: ABD EBC)

)

1) AD ║ EC , B is the mdpt of CD

- Given

2) A E, D C

2) Alternate Interior s

3) BD BC

3) Midpoint

4) ABD EBC

4) AAS

40

40

50

50

Why Angle-Angle-Angle (AAA)Doesn’t WorkThe angles are , but the sides are proportional.

(

A

D

F

(

C

B

Why Side-Side-Angle (SSA)Doesn’t WorkTwo different triangles can be formed if you use two sides and a non-included angle.

Theorem 4.8: Hypotenuse-Leg (HL) Theorem

- If the hypotenuse and a leg of a right are to a hypotenuse and a leg of a 2nd right , then the 2 ’s are

D

A

If BC EF and AC DF,

then ABC DEF

Special case of SSA

B

C

E

F

Then we can say: AB DE, A D , and C F

Reasons

Given: RS QT QRT is isosceles QRT is the vertex angleProve: QRS TRS1) RS QT, QRT is isosceles

- Given

2) QSR 90,

TSR 90

2) Definition of perpendicular

3) QSR TSR

3) Substitution

4) QR RT

4) Property of isosceles

5) RS RS

5) Reflexive

6) QRS TRS

6) HL

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