Geo 9 Ch 4.2-4.6 1

4.2 Congruency POWERPOINT Congruent figures have the same _______ and ________.

In a proof, if triangles are , and you plan to show that corresponding pieces are , use

"Corresponding parts of congruent triangles are congruent" or ____ ____ ____ ____ ____

To Prove : 1) Can the two triangles be proved congruent? If so, write the congruence and name the postulate used. If not, write “no”.

A

A A A

A

A A A

B

B B

B

B B

B B

C

C

C

C

C

C C C

D

D

D

D

D D

D

D

E

E

E

E E

F F F

X Y

Geo 9 Ch 4.2-4.6 2

Proofs : 1.

Given

ove

:

Pr :

OK bisects MOT

OM OT

MOK TOK

Statements Reasons

1. OK bisects MOT, OM OT 1.

2. 2. Def of bisector 3. 3. 4. MOK TOK 4. 2)

Given BA

BA

AYB AZB

: YZ

bisects YBZ

Prove:

Z

Y

B A

K

O

T M

Geo 9 Ch 4.2-4.6 3

3) PROVE: GHJ IJH

1. IHG IJG,

HJ bisects IHG and IJG

4) Prove: AMB CMB

1. AB BC , M midpoint of AC 1. Given

5) Prove: ABD CDB

1. ||

AB DC

AB DC 1. Given

H I

J G

4

3

2

1

A

B

C M

A B

C D

Geo 9 Ch 4.2-4.6 4

6) PROVE: ABC FED

1. || ,EF AB ||ED BC , AD FC 1. Given

PROVE: RST RWT

7)

1. 1 3, RS RW 1. GIVEN

F E

D

C

B A

4 3

2 1

S

T

W

R A

Geo 9 Ch 4.2-4.6 5

4-3 Using Congruent Triangles

Given OK: MK

KJ bisects MKO

Prove: KJ bisects MJO

Given AD BC

ove

:

Pr :

AD BC

AB CD

Given CD AB

ove CB

:

Pr :

D is the mp of AB

CA

Given P S

O

ove

:

Pr :

is the mp of PS

O is the mp of QR

O

J K

M

1

2

3

4

D C

B A

1

2

R

Q

O

S

P

D

C

B A

Geo 9 Ch 4.2-4.6 6

Given: ,

1 2,

NO OR SR OR

NO SR, Given:

1 4,

,PS RT NP NT

Prove: NP SP Prove: <PNR <TNS

Given: || , ||AB CD BC AD Given: ,

1 2

BC EF DC DE

Prove: B F Prove: 2 3

2 1

P R

S

O

N

4 3 2 1

T S R P

N

D

C B

A

2 1

G F

E

D

C

B

Geo 9 Ch 4.2-4.6 7

Given: 4 5,QR SR

Prove: AD BC Given: 3 4, RFI RIF

RA bisects <FRI, FR RI

Prove: FA IA

6

Q

5

4 3 2

1

S

R

P

6 5

4 3

2 1

A

I

R

F

Geo 9 Ch 4.2-4.6 8

4-4 The Isosceles Triangle Theorem SKETCHPAD By definition, ____________________________ _______________________________________ Drop an angle bisector from the vertex angle

Vertex

Legs Legs

Base

Base Angles

Base Angles

Geo 9 Ch 4.2-4.6 9

EXAMPLES 1. 2. 3. 4. ________ ______ ________ ________ 5. 6. 7. 8. __________ ________ __________ __________

9. Prove: secBDbi tsAC

1. 1 2

AB CB 1. Given

10.

PROVE: 1 2

1. ,BC DC BF DE 1. GIVEN

3x-2

16 65

x

5x

x+20

3x

x

80

x

x

x

100

x 50

x

2

1 D

C

B

A

F D

C

B

2 1

E

Geo 9 Ch 4.2-4.6 10

PROVE: X Z

11. 1. WY XZ XY YZ 1. GIVEN PROVE: <BAD = <BCD 12.

1. AB CB , AD CD 1. GIVEN 13.

1. AB AC 1. GIVEN

W

X Y

Z

4

3 2

1

D

C

B

A

E

D

C

B

A

2 1

Prove: 2B

Geo 9 Ch 4.2-4.6 11

PROVE: IJK is isosceles

12. 1. ||KM IJ 1 2 1. GIVEN

PROVE: BC DC

1. ||BD AE AC CE

13.

K

I J

M

1

2

D

E A

B

C

A

B

C D

E

F

G

90

Given a regular pentagon and a square. Find all the angles of the triangle.

Geo 9 Ch 4.2-4.6 12

4.5 AAS, HL

AAS: If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. Ex.

HL: If the hypotenuse and a leg of one right triangle are congruent to the correspondintg parts of another right triangle, then the triangles are congruent.

Ex. Are the triangles congruent?

Why is this true?

Given : 3 pieces of information 1) Right angles or triangles 2) congruent hypotenuses 3) 1 set of corresponding sides congruent.

Conclusion: the triangles are congruent!

A

B

C F D

E

Are the triangles congruent?

Geo 9 Ch 4.2-4.6 13

1. 2.

Given: YA AC ZC AC Given: BT TC , U O , AB CD

B is midpoint of AC Prove: UC BO

YB ZB

Prove: YA ZC

Z Y

C B A B A

C D

T

U O

2 1

Geo 9 Ch 4.2-4.6 14

3) 4)

Given: O p 90

MN QR OM=PQ

Prove: MOR QPN

O

M

N P

Q

R 1

2

B

A

D

C M

Given: <BAD and <BCD are right angles. AB BC

Prove: BAD BCD

Geo 9 Ch 4.2-4.6 15

5) 6)

1

1

D

A B

7

8 9

4

5 6

1 2

3

E C

F

1

0

1

1

1

2

Given: BE AD, AC BD, AC BE, DE EC

Prove: DEC is equilateral

J

G

H K M

N P

1 2

Given: K is midpt of HM, GH NM, HJ MP

HJ GK, MP KN

Prove: GK NK

Geo 9 Ch 4.2-4.6 16

7) 8)

P

R

H

J K M

O

Given:RK HR, JO PM, PH PM, PR PO

Prove:RK JO

A

B

C D E F

G

Given:BD CF, GE CF, CE DF, BC GF

Prove: ACF is isosceles

Geo 9 Ch 4.2-4.6 17

9) 10)

A B

C D

E

Given: AD BC, <DAB <CBA

Prove: ABE is isosceles

P

R

H

J K M

O

Given: PH PM, HJ KM, R and O are midpoints

Prove: RK OJ

Geo 9 Ch 4.2-4.6 18

4.5-4.6 OVERLAPPING TRIANGLES

WAYS THAT TRIANGLES OVERLAP: What do they share?

1)

A

B C

D

E

Given: AB DC AC DB

Prove: ABC DCB

A

B C

D

E

A

B C D E 1 2 3 4

5 6

Geo 9 Ch 4.2-4.6 19

2) 3)

11

D

A B

7

8 9

4

5 6

1

2

3

E C

F

1

0

11 12

Given: AD BD <1 3

Prove: ADC BDE

1

1

D

A B

7

8 9

4

5 6

1

2

3

E C

F

1

0

11 12

Given: 1 3, <2 4

Prove: AE BC

Geo 9 Ch 4.2-4.6 20

What do the triangles share? 4)

1

J

K G

M

H

3 1

2 Given: JH KH HG HM <1 3

Prove: JG KM

Geo 9 Ch 4.2-4.6 21

4.6 Using more than one pair of congruent triangles

1. 1. Given

A

B

O C

D

D

A O

C X

B

Given: AD AB DC BC

Prove: O is the midpoint of DB

:

Prove :

Given Marked

DC AB

Geo 9 Ch 4.2-4.6 22

:

Prove :

Given Marked

DC AB

C

A B O

D X

1

2

Geo 9 Ch 4.2-4.6 23

4.7 MEDIANS, ALTITUDES, PERPENDICULAR BISECTORS SKETCHPAD ..\..\..\3 SKETCHPAD FOR GEOMETRY\CH 4\4.7 Med, Alt, Perp Bis, Dist.gsp

DEF: MEDIAN________________________________________________________

_____________________________________________________________________

DEF: ALTITUDE______________________________________________________

_____________________________________________________________________

DEF: PERPENDICULAR BISECTOR______________________________________

TH 4-5: If a point lies on the perpendicular bisector of a segment, then_________________________________________________________ TH: 4-6: If a point is equidistant from the endpoints of a segment, then ________________________________________________________: TH 4-7: If a point lies on the bisector of an angle, then __________________________________________________________ TH 4-8: If a point is equidistant from the sides of an angle, then____________________________________________________________

Geo 9 Ch 4.2-4.6 24

Complete according to the picture.

1. If AB = BC, then ___________ is a median of APC.

2. If PC is a perpendicular bisector of ___________

then BC = DC.

3. If <APD is a right angle, then ________and __________

are altitudes.

4. If PC is a median of PBD, then ____________________

is a perpendicular bisector of ______________.

5. If BC = CD and PC BD , then __________ is a perpendicular bisector of _________.

6. If PC and AC are both altitudes of PCA , then __________is a right angle.

1. If BX bisects <ABC , then <_______ <______ and DX = ______.

2. If DX is the perpendicular bisector of EB , then ED = ______ and XE = _________.

3. If XB = XF, then ________ is the perpendicular bisector of BF , and < XBF _________.

4. If XD = XG, then ___________is the bisector of <________________.

D C B A

P

A

B C

D

E

F G

X

Geo 9 Ch 4.2-4.6 25

T

P S

L R

X

GIVEN:

RTS is isosceles. TX is the median to SR.

XL TR, XP TS

PROVE: XL XP

T

L S D

P

Given : PS AND LT are altitudes.

PT LS

Pr ove : <SPL TLP

Geo 9 Ch 4.2-4.6 26

Geo 9 Ch 4.2-4.6 27

Ch 4.5 Geometry Practice Problems

(1)

Given: AC BD

1 2

Prove: 13 14

(2)

Given: BE CD

15 16

4 6

Prove: 1 2

D

E F

9 10

11

6

7 8 3

4

5

G

1 2

A

B

C

13 14

12

15 16

5 6

13

14

1

2

3 4

C

D

A

B

7

8

9

10 12

11 E

Geo 9 Ch 4.2-4.6 28

Geo 9 Ch 4 Geometry Practice Problems

(1)

Given: AB CD

AF CE

AB = CD

Prove: AE = CF

(2)

Given: AD = AE

11 13

Prove: 7 8

A B

C D

E

F

1 3

2

4 5

6 7 8

9 10

11 12 13

14

15 16

17 18

A

B C

D E

F

1 3

2 4

5 6

7 8

9 10

11

12

13

14

Geo 9 Ch 4.2-4.6 29

Geo 9 Ch 4 Geometry Review Worksheet

(1) (2)

Given: 1 2 , 3 4 Given: AB BC , AD CD , AB = AD

Prove: AC = BD Prove: CA bisects BCD

(3) (4)

Given: AB = AC , D & E are midpoints Given: DF = EF , 2 4

Prove: BE = CD Prove: AD = AE

A B

5

4 2

3

E

1

C D

6

A

B

C

D

2

1 3

4

A

B C

7

8

9

4

5 6

1

2

3

D E

F

10

11

A

B C

7

8 9

4

5 6

1

2

3

D E

F

10

11 12

Geo 9 Ch 4.2-4.6 30

(5) (6)

Given: DF = EG , 15 16 , 2 4 Given: AF BC , DE BC

AC = BD , AF = DE

Prove: BD = EC Prove: AB = CD

(7) (8)

Given: 2 4 , 5 7 Given: AD = BE , CD = CE

Prove: ACD BDC Prove: AF = BF

A

B C

7 8 9

4

5 6

1 2

3

D E F

10

G

11 12

13 14 15 16

E

F 1

2

A B

C D

3 4

5 6

7 8

9 10

11 12

A B

C D

E

1 2

3

4

5 6

7 8

9 10

A B

C

D

7 8 9 10

3

4

5

6

E

F

1 2

Geo 9 Ch 4.2-4.6 31

Answers:

1.

Given

AAP

Subst

Reflexive

ASA

CPCTC

2.

Given

Def of Perp

Def of Rt. Triangle

Reflexive

HL

CPCTC Def of angle bisector

3.

Given

Def of Midpoint

SAP

Subst

Subst

Division Prop

Reflexive

SAS

CPCTC

4.

Given

VAT

ITT

SAS

CPCTC

Add Prop

SAP

Subst

Reflexive

AAS

CPCTC

5.

Given

AICP

PCAC

CITT

Reflexive

Add prop

SAP

Geo 9 Ch 4.2-4.6 32

Subst

SAS

CPCTC (ad=ae)

Add Prop

Subst

Subtract

SAS

CPCTC (5=6)

CITT

SAS

CPCTC

6.

Given

Def of Rt. Triangle

HL

CPCTC

AICP

If 1 pair of opp sides congruent and parallel then it’s a parallelogram

Opp sides of a parallelogram are congruent

7.

Given

CITT

VAT

SAS

CPCTC

Add prop

AAP

Subst

8.

Given

Add

SAP

Subst

Reflexive

SAS

CPCTC

VAT

AAS

CPCTC

Geo 9 Ch 4.2-4.6 33

Congruent Triangles More Review

1. Given: ACand BD bisect each other

Prove: ||AB CD

2. Given: AB = AC, ||DE BC

Prove: ADE is isosceles 3. Given: 5 6 , 3 4

Prove: DB = CE 4. Given: <8 = <10, DF = EF Prove: <11 = <12

A B

C D

E

1 2

3 4

5 6

7 8

B

A

D

C

E

A

B C D E 1 2 3 4

5 6

A

B C

7

8

9

4

5 6

1

2

3

D E

F

10

11 12

Geo 9 Ch 4.2-4.6 34

CONGRUENT TRIANGLES MORE REVIEW ANSWERS 1. GIVEN

AEB CED

< 3 2

AB || CD

2. GIVEN

B C

B ADE, <C AED

ADE AED

AD AE

ADE IS ISOSCELES

3. . GIVEN

1 3 180, <2+<4=180

<1SUPP<3, <2SUPP<4

1 2

AD AE

ADB AEC

DB CE

DB CE

4 GIVEN

7 8 180, <9+<10=180

<7SUPP<8 <9SUPP<10

<7 9

5 6

DF EF

DFB EFC

1 3, BF CF,

2 4

2 4, <1=<3

2 1 3 4

2 1 ABC, <3+<4=<ACB

<ABC=<ACB

<ABC ACB

ABC 11, <ACB 12

11 12

11 12

Geo 9 Ch 4.2-4.6 35

Geometry Extra Credit

Given: AC = DF , AB = DE , AG = DH , CG = GB , FH = EH

Prove: ABC DEF

B A

C

G

1 2 3

4

5 6

D E

F

H

7 8 9

10

11 12

Geo 9 Ch 4.2-4.6 36

SUPPLEMENTARY PROBLEMS CH 4

1. Is there anything wrong with the figure at the right?

2. Let a = (-5,0), B = (5,0), and C = (2,6); let K = (5,-2), L = (13,4), and M = (7,7). Verify that the length of each side of triangle ABC matches the length of a side of triangle KLM. Because of this data, it is natural to regard the triangles as being in some sense equivalent. It is customary to call the triangles congruent. The basis used for this judgment is called the side-side-side (SSS) criterion. If two figures are congruent, then their parts correspond. (Corresponding parts of congruent triangles are congruent - CPCTC) In other words, each part of one figure has been matched with a definite part of the other figure. In the triangles, which sides correspond? Measure the angles. Which angles correspond?

(DO #3-5 AT HOME AND BRING IN TRIANGLES FOR COMPARISON WITH YOUR GROUP) 3. Using a ruler and protractor, draw a triangle that has an 8cm side and a 6cm side, which make a

30 angle. This is a side-angle-side (SAS) description. Cut out the figure so that you can compare triangles in class. Will your triangle be congruent to those of your classmates?

4. With the aid of a ruler and protractor, draw a triangle that has an 8cm side, a 6 cm side, and a 45

angle that is not formed by the two given sides. This is a side-side-angle description. Cut out your figure so that you can compare triangles in class. Do you expect your triangle to be congruent with your classmates? 5. With the aid of a ruler and protractor, draw and cut out three non-congruent triangles, each of

which has a 40 angle, and a 60 angle, and an 8cm side. One of your triangles should have an angle-side-angle (ASA) description, while the other two have an angle-angle-side (AAS) descriptions. What happens when you compare your triangles with those of your classmates?

6. Suppose that triangle ACT has been shown to be congruent to triangle ION, with vertices A, C,

and T corresponding to vertices I, O,and N, respectively. It is customary to record this result by writing ACT ION. Notice that corresponding vertices occupy corresponding positions in the equation. Let B = (5,2), A =(-1,3), G = (-5,-2), E = (1,-3), and L = (0,0).

Using only these five labels, find as many pairs of congruent triangles as you can, and express the congruencies accurately.

7. How many ways are there of arranging the six letters of ACT ION to express the two-

triangle congruence?

6 8 not here

Geo 9 Ch 4.2-4.6 37

8. What can be concluded if it is given that a) ABC ACB b) ABC BCA CAB? 9. The parts of two triangles can be matched so that two angles of one triangle are congruent to the

corresponding angles of the other, and so that a side of one triangle is congruent to the corresponding side of the other, then the triangles must be congruent. Justify this angle-angle-corresponding side (AAS) criterion for congruence. Would AAS be a valid test for congruence if the word corresponding were left out of the definition? Explain.

9. A triangle that has two sides of equal length is called isosceles. Make up an example of an

isosceles triangle, one of whose vertices is (3,5). Give the coordinates of the other two vertices. If you can, find a triangle that does not have any horizontal or vertical sides.

10. Prove that in an isosceles triangle ( a triangle where two sides are congruent), the base angles

(the angles opposite the congruent sides) are congruent. To help, draw an auxiliary line from the vertex between the congruent sides, bisecting the angle and intersecting the opposite side.

11. Let A = (0,0), B = (4,3), C = (2,4), P = (0,4), and Q = (-2,4). Decide whether angles BAC and

PAQ are the same size (congruent) and give your reasons.

12. Plot the points A = (-4,0) and B = (4,0). Plot any point on the y-axis and call it P. What can you say about that point and the triangle it creates with A and B.

13. Give an example of a point that is the same distance from (3,0) as it is from (7,0). Find lots of

examples. Describe the configuration of all such points. In particular how does this configuration relate to the two given points?

14. Compare your example with those in your group.

a) What can you say about about all of the points that lie on the perpendicular bisector of a segment?

b) What can you say about the bisector of the vertex angle in an isosceles triangle? ?14. Let A = (-2,3) B = (6,7), and C = (-1,6). a) Find an equation for the perpendicular bisector of segment AB. b) Find an equation for the perpendicular bisector of segment BC. c) Find coordinates for a point K that is equidistant from A, B, and C. 15. Triangle ABC is isosceles, with AB congruent to AC. Extend segment BA to a point T (in other

words, A should be between B and T). Explain why <TAC must be twice the size of <ABC. 16. Given triangle ABC is isosceles, with AB congruent to AC. Extend segment AB to a point P

(where B is between A and P) so that BP is the same length as BC. In the resulting triangle APC, show that angle ACP is exactly three times the size of angle APC.

Geo 9 Ch 4.2-4.6 38

17. Simplify the equation 2222 )1()7()5()3( yxyx . Interpret your result.

(Challenge) 18. A segment that joins one of the vertices of a triangle to the midpoint of the opposite side is called

a median. Consider the triangle defined by A = (–2,0), B = (6,0), and C = (4,6). (a) Find an equation for the median drawn from A to BC. (b) Find an equation for the median drawn from B to AC. (c) Find an equation for the median drawn from C to AB. (d) Show that the three medians are concurrent, by finding coordinates for their common point. The point of concurrence is called the centroid of triangle ABC.

19. An altitude of a triangle is a segment that joins one of the three vertices to a point on the opposite

side, the intersection being perpendicular. In some triangles, it may be necessary to extend the side to meet the altitude. Now let A = (0,0), B = (10,0), and C = (4,12). (a) Find the length of the altitude from C to AB. (b) Find an equation for the line that contains the altitude from A to BC.

20. Given the triangle PQR with the points P = (-3, 6), Q = (2,0), and R = (6,0). (a) Find the altitude to segment QR from the point P. (b) Find the equation of the altitude to segment QR 21. There are four special types of lines associated with triangles: Medians, perpendicular

bisectors, altitudes, and angle bisectors. (a) Which of these lines must go through the vertices of the triangle? (b) Is it possible for a median to also be an altitude? Explain. (c) Is it possible for an altitude to also be an angle bisector? Explain.

22. Given a triangle, the point where two medians intersect (the centroid) is twice as far from one end of a median as it is from the other end of the same median. Triangle PQR is isosceles, with PQ = 13 = PR and QR = 10. Find the distance from P to the centroid of PQR. Find the distance from Q to the centroid of PQR.

Geo 9 Ch 4.2-4.6 39

Geo 9 Ch 4.2-4.6 40