Geometry Chapter 4 Triangle Congruence

Proofs

1

Name: _________________________________________________________________ Geometry Assignments – Chapter 4: Congruent Triangles

Date Due

Section

Topics

Assignment

Written Exercises

NOTE: We will be using

Two-Column Proofs

4.1

Congruent Figures Corresponding Parts

Pg. 120-121 #1-7, 10, 11, 20, 21

4.2

4.5

Ways of Proving ’s -SSS Postulate

-SAS Postulate

-ASA Postulate

-AAS Theorem

-HL Theorem

Pg. 124-125 (bottom) #1-15

Pg. 142 (Classroom Exercises) #1-13

Worksheet

Triangle

Congruence

Proofs

Practice

Pg 125-126 #17-19

Pg 143-144 #1-4

STUDY for QUIZ Next Class on

Sections 4-1, 4-2, and 4-5

4.3

Corresponding Parts of

’s are

Pg. 130-131 #1, 3, 4, 7, 8, 10, 11

2

4.4

Isosceles Triangles and their Theorems/Corollaries

Linear Pair

Pg. 137-139 #1-8, 13, 14, 16, 28

4.6

4.7

Using more than One Pair of

’s

Def. of Median Def. of Altitude Def. of Bisector Distance from a Point to a

Line

Pg. 149 #7-9, 11, 12

Triangle Proof #1 WS - #6, 12

Proof Worksheet with Median,

Altitude, Perpendicular Bisector

Chapter 4

Review Suggested Chapter 4 Review

Questions from your Textbook and Notes/Classwork

Pg. 132-133 (Self-Test 1 #1-8) Pg.146 (Self-Test 2) #1-5

Pg. 159 (Self Test 3) #1-6

Pg 160-161 (Chapter Review)

#1-8, 11-20

Pg 162-163 (Chapter Test) #1-19

PLUS…any questions from worksheets used in class/HW that we did not complete.

All answers for above questions are available, simply ask for them!

STUDY MATH BY DOING THE MATH!!!

Use Suggested Practice as a Guide,

Ask for help!!!!

3

Sect. 2.1

4

Section 4.2 – Ways of proving triangles congruent Date: ____________

SSS Postulate -

SAS Postulate -

ASA Postulate -

Section 4.5 – More ways of proving triangles congruent

AAS Theorem -

HL Theorem -

5

Triangle Congruence WS 1 Date: ____________

The triangles have been marked to show congruent sides or angles. You may mark the

triangles to show the reflexive property if needed. Determine the following:

6

Triangle Congruence WS 2 Date: ____________

******Remember to mark vertical angles or the reflexive property if needed!******

Part 2:

7

Part 2 continued:

Part 3: Determine whether the following pairs of triangles are congruent. If congruent, state the reason and

give a triangle congruence statement [ by ASAWXY WXZ ]. If there is not enough information given,

state “not enough info”.

8

A

D

B

C E

A B E

C

1 2

3 4

D

4-2 Proof Notes Date ______________

1. Given: CDand AB intersect at E.

BA bisects CD , ;AC CD BD CD

Prove: ACE BDE

Statements Reasons

1. andCD AB intersect at E 1.

2. 2. vertical angles are

3. BA bisects CD 3.

4. E is the midpoint of CD 4.

5. CE DE 5.

6. ;AC CD BD CD 6.

7. 7. right ’s

8. 8. all right ’s are .

9. ACE BDE 9.

2. Given: AB bisects CAD

CBE DBE

Prove: ACB ADB

Statements Reasons

1. AB bisects CAD 1.

2. 1 2 2.

3. AB AB 3.

4. CBE DBE 4.

5. 3 & CBE suppl.

4 & DBE suppl.

5.

6. 6. supplements of ’s are

7. ACB ADB

9

Q M

P

R

A B

X

D C

1

2

3. Given: RP RQ

M is the midpoint of PQ

Prove: PRM QRM

1. RP RQ 1.

2. M is the midpt. of PQ 2.

3. 3. midpt segs

4. RM RM 4.

5. PRM QRM 5.

4. Given: A C

X is the midpoint AC

Prove: ABX CDX

Statements Reasons

1. 1.Given

2. 2.Given

3. AX CX 3.

4. 4. Vertical ’s are .

5. 5.

10

A

C

D B

1 2

P

R

S Q 1 2

5. Given:

;RS PQ

PR QR

Prove: PRS QRS

Statements Reasons

1. 1.Given

2. 2. lines form right ’s.

3. 3. Triangle with a right is a right triangle.

4. 4. Reflexive Property

5. PRS QRS 5.

6. Given: CD bisects ACB

AC BC

Prove: ACD BCD

Statements Reasons

1. 1. Given

2. 2. Given

3. 1 2 3.

4. 4.

5. 5. S.A.S. Postulate

11

J

M L

K 2 1

7. Given: MK LJ

1 2

Prove: MJK LKJ

Statements Reasons

1. 1. Given

2. 2. Given

3. JK KJ 3.

4. 4.

12

Beginning Triangle Proofs Date: ___________

1.

2.

13

3.

4.

Given //AB CD , E is Midpoint of AC .

Prove AEB CED

14

5. Given JL MK and ML LK .

Prove: JLM JLK

6. Given AD bisects BAW and BDW

Prove BAD WAD

15

Section 4-3: Using Congruent Triangles Date ____________

Recall – If two triangles are congruent then the corresponding angles are congruent and the corresponding sides

are congruent.

To use in proof : Corresponding Parts of 's

1. Given: ;XW XZ WY ZY 2. Given: ;RY RX PR QR

Prove: XWY XZY Prove: P Q

________________________________________________________________________

3.

________________________________________________________________________

4.

X

W

Y V

Z

5 6

7 8

R

X Y

P Q

16

5.

________________________________________________________________________

6.

17

_______________________________________________________________________________

4-4 Notes: Isosceles Triangle Thm.

Isos. Δ Thm.  In a Δ, 2 sides ≅ ⇒<s opposite are ≅

Converse isos. Δ thm. In a Δ, 2<s ≅ ⇒ sides opposite are ≅

________________________________________________________________________________

18

A

B

C

D

P 2

1

A B

C D

X

1 2 3

4

5 6

Triangle Proofs #1 WS Date _________________

1.

2. Recall:

________________________________________________________________________

3. Given: bisects ;AB CD C D 4. Given: ;AX BX CX DX

Prove: ACP BDP Prove: ACX BDX

5. Given: ;AC BC AX BX 6. Given: ; ;WY ZY W Z

VYW VYZ

Prove: ACX BCX Prove: XWY XZY

B. AXD BXD

A B

C

X

1 2

3 4 5

6 7

8

D

X

W

Y V

Z

5 6

7 8

19

A

D

B

C

P

A B

C D

X

1 2 3

4

5 6

7. Given: 1 2; AX BX 8. Given: ;RZ WZ RT WS

Prove: AC BD Prove: is isoscelesSTZ

9. Given: ;AP DP BP CP 10. Given: ; ;PR RQ PS SQ

RQ SQ

Prove: AB DC Prove: PRQ PSQ

11. Given: ; ;AC BC BD AD

AC BD

12. Given:

; ;EFA CFB FA DE

FB DC

Prove: ABC BAD Prove: DAF DBF

C D

A B

X

1 2

3 4

R S T W

Z

R

P Q

S

A B

C

D

E 1

2 3 4

5

F

6

20

13. Given:; ;

midpoint of

A B PQ AB

Q AB

14. Given:

; ;AQ PQ BQ PQ

AP BP

Prove: APQ BPQ Prove: APQ BPQ

_______________________________________________________________________

15. Given: ;ML NL PNM QMN

Prove: MPN NQM

A B

P

1

Q

2

4 3 A B

P

1

Q

2

4 3

L

P Q

M N

21

Def. Linear Pair- Two angles form a linear

pair iff they are adjacent and the non-

common rays form a straight line.

_________________________________________________________________________________

1.

Prove: BAC DAC

Proofs with Multiple Δ's

2.

3.

22

4-6 Notes WS: Using More than one pair of congruent Triangles Date: ____________

1.

_________________________________________________________________

2.

______________________________________________________________________

3.

______________________________________________________________________

4.

23

5.

_______________________________________________________________________

6.

24

Given altitude in proof:

Altitude

' whicharert s

Given median in proof:

median mdpt

mdpt segs

Section 4-7: Median, Altitude, Perpendicular Bisectors in Triangles Date: ___________

Altitude:

A segment from the vertex to a point on the opposite side such that the segment is perpendicular

to the side of the triangle.

To prove altitude, do steps in reverse!

_________________________________________________________

Median:

A segment from the vertex to the midpoint of the opposite side of the triangle.

To prove median, do steps in reverse!

_________________________________________________________

Perpendicular Bisector:

A line or segment perpendicular to a side of the triangle and passing through the midpoint of the

side.

We will not use perpendicular bisector in proofs!

25

C

A BD E F

S T

R

K

1. In RST , if U is the midpoint of RS , then TU is a(n) _______________ of RST .

2. Given: ; ;CD AB AF BF ACE BCE , name

(a) an altitude of ABC __________________

(b) a median of ABC __________________

(c) an angle bisector __________________

3. If R is the perpendicular bisector of ST , then

R is equidistant from _____ and _____.

Thus _____ _____ .

4. Perpendicular Bisector 5. Median

6. Altitude

26

O

A

D

B

C

Overlapping Segments Theorem-

Given a segment with points A, B, C and D arranged as shown, the following statements are

true:

If AB CD , then AC BD

or

If AC BD , then AB CD

*Warning – You must have a step in proof that states BC BC first!

Example

Given:

;

;

DE FC

AF EB

EG FG

Prove: AD BC

Overlapping Angles Theorem-

Given AOD with points B and C in the interior as shown, the following statements are true:

If AOB COD , then AOC BOD

or

If AOC BOD , then AOB COD

*Warning – You must have a step in the proof that states BOC BOC first!

A DB C

27

Example:

Given: ;

1 2

RS RV

Prove: ST VU

In class practice:

Given: AD is a median and altitude of ASW

Prove: ASW is isosceles

28

Now for some Algebra Review

1. Given AB CD , AB = 2x + 3, AC = 10x – 14,

and BD = 7x + 0.4. Find x and CD.

2. Given AC BD , AD BE , AE = 416,

and BC is 1 more than twice CD. Find AB.

3. Given AVB CVD , 5 73m AVC x ,

10 83m BVD x , and 33m AVB x .

Find x and m AVB .

4. Given AVD BVE , BVC CVE ,

88m AVE , and m CVD is two less than double m AVB .

Find x and m AVC .

29

4-7 Homework WS Date:__________________

1. Given: ;ABC BD is both the median and the altitude to AC

Prove: AB CB

_______________________________________________________________________

2. Given: PM is median of PQR ; Q R

Prove: PM is bisector of QR

________________________________________________________________________

3. Given: QKR is isosceles with QK KR

KS is a median of QKR

Prove: TM MV

_______________________________________________________________________

4. Given: AC is the median to BE 5.

EC is the median to AD

Prove: AB ED

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6. 7.

_________________________________________________________________

8. Given: S is the midpoint of PR ;

QS is the altitude from Q to PR

Prove: PQS RSQ

Now for some overlapping segment and angle theorem.

1. Given: ;MP QN KM KN 2. Given: ; ;A B AD BE

ADG BEF

Prove: MKQ NKP Prove: AFE BGD

M P Q N

K

A D E

B

G

C

F

1 2 3 4

31

3. Given: ; ;

;

K Q MK MQ

PM MK LM MQ

4. Given:

; ;

;

AB AC BG CE

DE BC FG BC

Prove: L P Prove: BD CF

K

M

Q

L

P

3 2

1

A

B C

D

E

F

G

32

PROOF REVIEW CLASSWORK

1. Given: ;AR ER

EC AC

2. Given:

;SA NE

SE NA

Prove: A E Prove: SA NE

________________________________________________________________________

3. Given: ;

is midpoint of

E T

M TE

4. Given:

;MN MA

ME MR

Prove: MI MR Prove: E R

________________________________________________________________________

5. Given: ;EM IT

EI MT

Prove: E T

R

E

C

A

E N

A S

I

T

M

R

E

Y

E

N

M A R

E M

T I

33

Review Chapter 4 Classwork Date: ________

1. Given: 2 4; midpoint ofA BC

Prove: bisectsAD BDC

_______________________________________________________________________

2. Given: midpoint ofF WB and AS

Prove: AB WS

3. Given: Altitude EF of Isosceles ∆EDW

Prove: EF is a median

4. Given: GE AE , GD AD , GE GD , GHB CHG

Prove: BH CH

34

P

Q

R

S

O

5. Given: PQ is an altitude of PQR , PS is an altitude of PSR ,

PQ PS

Prove: PO is a median of PQS

6. Given: .Cismidpt AE , ||AB DE

Prove: bisectsAE BD

7. Given: Z C , FZ WC , AR bisects FAW

Prove: AR is median to Triangle FAW

8. Given: OT TC ; ER BR ;

TC BR ; OT ER

Prove: CO BE

35

9.

Given:

1 3

AB AE

B E

Prove: BC DE

_____________________________________________________________

10. Given: BE CD

BD CE

Prove: ABC is isosceles

A

E

C D

B

1 2 3

A

B C

D E

36

Chapter 4 Review Homework Date: ____________

1.. Given:

altitude of ;

altitude of ;

1 2;

BE ABC

CE DCB

AB DC

Prove: ABE DCE

__________________________________________________________________

2. Given: sec ;AD bi ts BAC

AB AC

Prove: 1) ;

2) median of

BED CED

ED BED

____________________________________________________________________

3. Given: ;BDE CDE

B C

Prove: bisectsAD BAC

_____________________________________________________________________

4. Given: ; 1 2CA CB

Prove: CDE CED

A

B C

D

E

2 1

A

B C D

E

1 2

A

B

C

D E 2

1

A

C

E D

1

B

2

37

5. Given: ;HA HF HM HQ

Prove: FM AQ

_____________________________________________________________________

6. Given: ; ;PQ PR PS PT QV RW

Prove: SV TW

______________________________________________________________________

7. Given: ; ;AD EB AC BC AFE BGD

Prove: AFE BGD

_______________________________________________________________________

8. Given: ;VW WX VY XZ

Prove: is isoscelesWYZ

H

M Q

A F

B

A D E

B

G

C

F

P

Q R

S

V

T

W

Z X Y V

W

38

9. Given: ; ;

1 4

RS XY RS PQ

Prove: PRS QRS

_____________________________________________________________________

10. Given: ; ;DG DH AG BH

AC BC

Prove: AE BF

_______________________________________________________________________

11. Given:

bisects ;

bisects ;

AB CD

CD AB

AC BD

Prove: ACP BDP

P R

Q 5 6

1

2 3

4

X S Y

A B

C

E

1

F

2

4 3

G H

D

A

B

C

D

P 2

1