Geometry Workbook 5

Geometry Workbook 5:

Definition of Congruence, Triangle Congruence, Proofs, CPCTC,

Triangle Relationships

Student Name __________________________________________

STANDARDS:

G.CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G.CO.B.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.

G.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motion.

G.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

SKILLS:

I will be able to show two figures are congruent if there is a sequence of rigid motions that map

one figure to another.

I will be able to show that two figures are congruent if and only if they have the same shape and

size.

I will be able to use composite transformations to map one figure onto another.

I will be able to recognize the effects of rigid motion on orientation and location of a figure.

I will be able to use the definition of congruence as a test to see if two figures are congruent.

I will be able to identify corresponding angles and sides based on congruence statements.

I will be able to develop and write congruence statements for two congruent triangles.

I will be able to determine if two triangles are congruent based on their corresponding parts.

I will be able to explain and apply the criteria of SSS, SAS, ASA, AAS, and HL to prove triangle

congruency.

I will be able to explain in what cases AA and ASS do and don't prove triangle congruency.

I will be able to prove and apply that the sum of the interior angles of a triangle is 180°.

I will be able to prove and apply that the base angles of an isosceles triangle are congruent.

(HONORS) I will be able to prove and apply the midsegment (midline) of triangle theorem.

(HONORS) I will be able to prove that the medians of a triangle meet at a point, a point of

concurrency.

(HONORS) I will be able to prove and apply the exterior angle theorem.

(HONORS) I will be able to determine the conditions for forming a triangle, when given three

lengths.

Notes:

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

G.CO.B.6 GUIDED PRACTICE WS #1 – geometrycommoncore.com 1

1. Rigid motions are:

2. Figures are congruent if

3. List the isometric transformations: ________________________________

A congruence statement relates one identical object to another by identifying the corresponding parts that match each other.

Quad ABCD Quad MNOP 4. List the Congruent Angles 5. List the Congruent Sides

NYTS (Now You Try Some)

6. Determine the congruent sides and angles from the congruence statement.

a) AGR HJM b) Quad HJKM Quad WSRT

CPCFC – Corresponding Parts of Congruent Figures are Congruent. A translation (x, y) ---> (x + 6, y - 3)

maps these two quadrilaterals,

so Quad QRST Quad UVWX

A rotation of 270 about the origin

maps these two quadrilaterals,

so Quad QRST Quad UVWX

A reflection over the y axis

maps these two pentagons,

so ABCDE JMWYH

T S

R

Q

U

V

WX

U

V

W

X

T

S

R

Q

J

HY

W

MB

C

DE

A



7. Name the transformation or sequence of transformations that maps one figure onto the other. Then complete the congruence statement.

a)

b)

c)

Transformation(s):

______________

ABC

Transformation(s):

________________________

Quad ABCDQuad

Transformation(s):

________________________

Quad ABCDQuad


8. Is Quad QRST Quad PLKJ? Determine a sequence of isometric transformations from QRST to PLKJ (name it specifically and also graph it)

Original Relationship

______________________

______________________

H

L

K

A

B

C

R

S

P

M

A

B

C

D

L

R

Y

P

A

B

C

D

K

J

L

P

T

S

R

Q

K

J

L

P

T

S

R

Q

K

J

L

P

T

S

R

Q

G.CO.B.6 WORKSHEET 1 – geometrycommoncore.com NAME: _______________________ 1

1. Quadrilateral ABCD is congruent Quadrilateral HJKL. Complete the following congruence statements.

A ______ JK ______ L ______ DA ______

2. Pentagon ABCDE is congruent to Pentagon HJKLP. Complete the following congruence statements.

B ______ KL ______

J ______ DE ______

3. ABC is congruent to another triangle. Provided is some information about the two triangles, AB LP

and CA ML . From this information determine the triangle congruence statement.

ABC _________ 4. Determine the missing information.

ABC DEF QUAD ABCD QUAD MNOP QUAD ABCD QUAD AMCG

mB = ________ AC = ________

mF = ________ DE = ________

mM = _______ OP = ________

mC = ________ AD = ________

mDAB = ______ MC = _______

mBCD = ______ AM = _______


Transformations:

ABC _______

Transformations:

ABC _______

Transformations:

ABC _______

P L

K

J

H

A

BC

D

E

93°

7.6 cm

73°5.4 cm

F

E

D

A

B

C 112°

87°72°

4.0 cm

3.8 cm

4.6 cm

3.7 cm

O

N

M

P

D

A

B

C 28°

81°

62°

10 cm13 cm

16 cm

12 cm

M

G

D

A B

C

J

N

P

A

B

C

JN

P

A

B C

H

L

K

A

B

C


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.

1. Congruence of triangles is defined by:

A congruence statement for triangles

relates one identical object to another by identifying the corresponding parts that

match each other.

2. Determine the congruent sides and angles from the congruence statement.

ABC DEF List Congruent Angles

List

List Congruent Sides

CPCTC – Corresponding Parts of Congruent Triangles are Congruent.

3. TRY AXD 4. PLC MNB

Determine congruence using a single or sequence of isometric transformations. A reflection (x, y) --- > (x, -y) maps these two triangles,

A translation (x, y) --- > (x + 3, y - 5) maps these two triangles,

A rotation (x, y) --- > (-y, x) maps these two triangles,

5. Are the ’s congruent?

Yes or No

Create the statement.

_________ ________


Yes or No


_________ ________


Yes or No


_________ ________

DG

R

A

B

C

I

H

A

B

CGH

K

T

A

B

C


Is ABC KJL? Is there a sequence of isometric transformations that map one onto the other?

Original Relationship A reflection over the y axis A translation of <4,-4>

YES, ABC KJL is because I can map ABC onto KJL using a reflection and then a translation.

8. Is ABC DEF

Determine a sequence of isometric transformations from ABC to DEF (name it specifically and also graph it).

Original Relationship

______________________

______________________

9. Name the transformation or sequence of transformations that map one figure onto the other. Then complete the congruence statement.

Transformations: (Start with ABC)

A reflection over the ____________

Followed by

A translation of ____________

ABC _______

Transformations: (Start with ABC )


Followed by


ABC _______

L

J

A

B

C

K L

J

C'

B'

A'A

B

C

K L

J

C'

B'

A'A

B

C

K

F

EDA

BC F

EDA

BC F

EDA

BC

FE

D

A

BC

H G

T

A

BC

G.CO.B.7 WORKSHEET #1 – geometrycommoncore.com NAME: _______________________ 1

1. HYZ is congruent KLR. Complete the following congruence statements.

L ______ LR ______ H ______ ZH ______

2. ABC is congruent to TDJ. Complete the following congruence statements.

B ______ JD ______

T ______ BC ______

3. ABC is congruent to another triangle. Provided is some information

about the two triangles: AB AL and CA GA . From this information determine the triangle congruence statement.

ABC _________

4. Determine the missing information.

ABC TDJ AFY RQK CKB KCJ

mC = ________ TJ = ________

mB = ________ AB = ________

mK = _______ AF = ________

mA = ________ KR = ________

mJ = ______ BK = _______

mJCK= ______ JK = _______


Transformations: (Start with DEC)

A rotation about the origin of _______

Followed by


DEC _______

Transformations: (Start with FLT)


Followed by


FLT _______

J

D

T

A

B

C

4.2 cm50°

4.7 cm70°

J

T

D

B

A

C

3 cm

4 cm

53.1°

K Q

R

A

FY

7.3 cm6.6 cm

70°

79°

J

C

B

K

H F

E

C

D

G

L

F

E

O

D

T

G.CO.B.8 ACTIVITY #1 – geometrycommoncore.com 1

Components of Triangles

ed

c

ba

5

4

3

2

1

5

4

3

2

1


Our goal in this activity is to determine what the minimum requirements are to establish that two triangles are congruent to each other. Triangle congruence is determined if one triangle can be mapped onto another using only isometric transformations. In this activity we will be creating triangles based on certain requirements and then classmates will map their triangle onto each other’s to see if they are congruent. This is NOT a formal proof but we will be able to quickly get a sense of which criterion work and which ones don’t.

Testing Criteria for Triangle Congruence

CRITERIA #4 – GIVEN TWO ANGLES (AA)

Create a triangle using 1 and 2. To do this you will want to

copy 1 onto the piece of patty paper and then using the

provided sheet close the triangle using 2. Trace the completed triangle.

LABEL 1 AND 2 in the completed triangle.

Repeat again but use the Example #2 values

Example #1

1 and 2 (compare your with 3 other students)

Example #2

3 and 5 (compare your with 3 other students)

Do you think AA is a congruence criterion? Why or Why not?

CRITERIA #5 – GIVEN AN ANGLE, THE INCLUDED SIDE AND OTHER ANGLE (ASA)

Create a triangle using 3, side d and 1. The order of

these matter… side d must be the included side (the side directly

between 3 and 1). Trace the completed triangle.

LABEL 3 AND SIDE d AND 1 in the completed triangle.


Example #1

3 side d 1

(compare your with 3 other students) Example #2

4 side b 2 (compare your with 3 other students)

Do you think ASA is a congruence criterion? Why or Why not?

21

d

13

CRITERIA #1 – GIVEN ONE SIDE (S) CRITERIA #2 – GIVEN ONE ANGLE (A)

No need to investigate this one –

it is obvious that many different triangles

can be formed by only having one side in common.

No need to investigate this one –

it is obvious that many different triangles

can be formed by only having one angle in

common.

CRITERIAN #3 – GIVEN TWO SIDES (SS)

Using patty paper, create a triangle with

side lengths of a & c.

SIDE a and SIDE c

Why doesn’t SS have enough information to

guarantee congruence between the triangles?


CRITERIA #6 – GIVEN A SIDE, THE INCLUDED ANGLE AND OTHER SIDE (SAS)

Create a triangle using Side c, 3 and Side d. The order of

these matter… 3 must be the included angle (the angle directly between Side c and Side d).

Trace the completed triangle.

LABEL SIDE c, 3 AND SIDE d in the completed triangle.


Example #1

Side c 3 Side d


Side b 2 Side c (compare your with 3 other students)

Do you think SAS is a congruence criterion? Why or Why not?

CRITERIA #7 – GIVEN A SIDE, THE INCLUDED ANGLE AND OTHER ANGLE (SAA/AAS)

Create a triangle using Side c, 1 and 2. Again order

matters, copy Side c and then place 1 at the one of the ends of

Side c….. then use 2 to close the triangle. LABEL SIDE c, 1 AND 2 in the completed triangle.


Example #1

Side c 1 2


Side b 3 4 (compare your with 3 other students)

Do you think SAA/AAS is a congruence criterion? Why or Why not?

CRITERIA #8 – GIVEN AN ANGLE, A SIDE AND THEN A SIDE (ASS/SSA)

Create a triangle using 2, Side c, and then Side b. This

one is a little tricky to form so let me try to help you….. Copy 2

– then use Side c as one of the sides of 2 and then connect Side b to other endpoint of Side c so that it closes the triangle.

LABEL 2, SIDE c, AND THEN SIDE b in the completed triangle.


Example #1

2 Side c Side b (compare your with 10 other students)

Example #2

1 Side a Side e (compare your with 10 other students)

Do you think ASS/SSA is a congruence criterion? Why or Why not?

c

d

3

2

c

1

c

2


CRITERIA #9 – GIVEN A SIDE, A SIDE AND A SIDE (SSS)

Copy Side a onto a piece of patty paper, then copy Side b onto a different piece of patty paper, and then copy Side c onto a

different piece of patty paper….. now overlap them until they close the triangle… copy that

triangle onto one of those sheets.

LABEL SIDE a, SIDE b AND THEN SIDE c in the completed triangle.


Example #1

Side a Side b Side c


Side b Side c Side d

(compare your with 3 other students)

Do you think SSS is a congruence criterion? Why or Why not?

Example #3

Side d Side a Side e What happened here? Why did this happen? What must be true for a triangle

to be formed?

CRITERIA #10 – GIVEN A SIDE, A SIDE AND A SIDE (AAA)

Create a triangle using 1, 3, and 5. This one is a little

tricky to form so let me try to help you….. Copy 1 onto a piece

of patty paper, then copy 3 onto a different piece of patty

paper, and then 5 onto a different piece of patty paper….. now overlap them until they close the triangle… copy that

triangle onto one of those sheets.

LABEL 1, 3, AND THEN 5.

Example #1

1 3 5

(compare your with 3 other students)

Do you think AAA is a congruence criterion? Why or Why not?

Why is AA the same as AAA?

Which criterion worked? Which criterion didn’t work?

c

b

a


1. Which criteria established congruent triangles?

2. Which criteria did not establish congruent triangles?

3. What are the AS1S2 cases?

CASE #1 – AS1S2, when S2 is greater than S1.

CASE #2 – AS1S2, when S2 is less than S1. (Too Short)

CASE #3 – AS1S2, when S2 is less than S1 (1 Intersection)

CASE #4 – AS1S2, when S2 is less than S1. (2 Intersections)

4. Are the following pairs of triangles congruent?

If they are, then name their congruence criteria. (SSS, SAS, ASA, AAS, HL or AS1S2 (S2 > S1))

a)

b)

c)

Yes / No

If Yes, Criteria is __________

Yes / No

If Yes, Criteria is ___________

Yes / No


R

E

W

A

B

C o

o

MU

Y

C

A

B

H

T

Y

D C

B


d)

e)

f)

Yes / No

If Yes, Criteria is _________

Yes / No


Yes / No


g)

h)

i)

Yes / No


Yes / No


Yes / No


j)

k)

l)

Yes / No


Yes / No


Yes / No


m)

n)

o)

Yes / No


Yes / No


Yes / No


p) q) r)

Yes / No


Yes / No


Yes / No


o

x

x

o

B

T

E

G

*

*x

x

U

L

K

T

E

G

*

*

o

o

x

x

H

X

W

C

E

D

o

o

L

K

T

A

B

CR

J

T

H

oo

H

Y

GA

C

B

oo

M

H

C

AB

o

o

T

GB

C

D

H

ox

x

oU

L

K

T

E

G

E

D

C

B

o

x

x

o

Y

D

T

E

G

o

o


5. Are the following pairs of triangles congruent? If YES, create a congruence statement and name the congruence criteria (SSS, SAS, ASA, AAS, HL or AS1S2 (S2 > S1)).

a) Yes / No

______ ______

Criteria ___________

b) Yes / No

______ ______


c) Yes / No

______ ______


d) Yes / No

______ ______


e) Yes / No

______ ______


x

xo

oN

M

E

D

C

x

o

x

o R

E

D

C

o

o T

E

D

C

o

17 cm

4 cm

17 cm4 cm

oH

L

KE

D

C

*

*

o

xo

xH

L

K

E

D

C

G.CO.B.8 WORKSHEET #1a – geometrycommoncore.com NAME: _____________________________ 1

1. Are the following pairs of triangles congruent? If they are, then name their congruence criteria.

(SSS, SAS, ASA, AAS, HL)

a) Yes / No __________ b) Yes / No __________ c) Yes / No __________ d) Yes / No __________

e) Yes / No __________ f) Yes / No __________ g) Yes / No __________ h) Yes / No __________

2. Are the following pairs of triangle congruent? If YES, create a congruence statement and name the

congruence criteria (SSS, SAS, ASA, AAS, HL).

a) Yes / No

_____ _____

Criteria ________

b) Yes / No

_____ _____

Criteria ________

c) Yes / No

_____ _____

Criteria ________

d) Yes / No

_____ _____

Criteria ________

e) Yes / No

_____ _____

Criteria ________

f) Yes / No

_____ _____

Criteria ________

g) Yes / No

_____ _____

Criteria ________

h) Yes / No

_____ _____

Criteria ________

i) Yes / No

_____ _____

Criteria ________

j) Yes / No

_____ _____

Criteria ________

o

o *o

o*

o

o

*

o

*

o**

o

o

P

GH

C

D

E

**

oo

D

A B

C

P

AJ

C

H

B

C

P

**

oo P

A

JC**

Y

T

J

K

G

RT

P

4 cm4 cm

E

D

C

B

A

o

o

E

B C

D

K

M

H

P

A

G.CO.B.8 WORKSHEET #1a – geometrycommoncore.com 2

3. Are the following pairs of triangle congruent? If YES, create a congruence statement and name the

congruence criteria (SSS, SAS, ASA, AAS, HL).

a) Yes / No

Criteria ________

b) Yes / No

Criteria ________

c) Yes / No

Criteria ________

d) Yes / No

Criteria ________

e) Yes / No

Criteria ________

f) Yes / No

Criteria ________

g) Yes / No

Criteria ________

h) Yes / No

Criteria ________

i) Yes / No

Criteria ________

j) Yes / No

Criteria ________

4. Jeff states that PLN CVB because of ASA. Nancy says that she knows something that would allow her to use AAS. What does she know that would allow her to use AAS for these triangles?

5. Why does HL (Hypotenuse – Leg) work as a triangle congruence criterion?

o

o HG

FI

TH

A

B

C

o

o

T

R

Q

S

o

o*

*L

UJ

AK

HE T

BCD

oo

J

D

E

F

oo

F

E

D

CA

B o

o

BG

A

C

D

R

F

G

K

PL

L

H

G

F

**

o

oP

L

N

B

C

V

F

E

D

C

B

A

G.CO.B.8 GUIDED PRACTICE WS #2/#3 – geometrycommoncore.com 1

In what ways have we proven some of the concepts so far?

A two column proof is another way to organize a proof.

GIVEN:

PROVE:

DIAGRAM:

STATEMENT REASON

A two column proof is just one way to organize a proof. Instead of explaining our reasoning and our

statements in a paragraph, we organize it here into columns.

In this objective we are trying to establish two triangles to be congruent so ultimately we need to show that

one of the congruence criterion has been satisfied (SSS, SAS, ASA, AAS, HL, AS1S2 (S2 > S1). Once you have

established that a criterion has been met, you are able to declare the congruence of the triangles involved.

Prove the following relationships.

1. GIVEN: BDT GTD &

BD GT

PROVE:

BTD GDT

2. GIVEN: B D &

AC EC

PROVE:

BCA DCE

STATEMENT REASON

STATEMENT REASON

TG

B

D

B

C

A

E

D

G.CO.B.8 GUIDED PRACTICE WS #2/#3 – geometrycommoncore.com 2

3. GIVEN:

AB AD & BC DC

PROVE:

ABC ADC

4. GIVEN: B D & C is the

midpoint of AE

PROVE:

BCA DCE

STATEMENT REASON

STATEMENT REASON

D

A C

B

D

CA

E

B

G.CO.B.8 WORKSHEET #2 – geometrycommoncore.com NAME: _______________________ 1

1. Prove the following relationships.

a) GIVEN:

B E & CB DE

PROVE:

EAD BAC

b) GIVEN:

VC DB & VB DC

PROVE:

BVC CDB

STATEMENT REASON

STATEMENT REASON

c) GIVEN:

B D & BC DC

PROVE:

ACB ECD

d) GIVEN:

AD CB & AB CD

PROVE:

ABD CDB

STATEMENT REASON

STATEMENT REASON

e) GIVEN:

AE BE & DE CE

PROVE:

AED BEC

f) GIVEN:

T is the midpoint of ME

& T is the midpoint of GJ

PROVE:

MGT EJT

STATEMENT REASON

STATEMENT REASON

A

B

E D

CDB

CV

C

A E

D

B

A

B

D

C

C

BE

A

D

T

M

E

J

G

G.CO.B.8 WORKSHEET #2 – geometrycommoncore.com 2

g) GIVEN:

1 2 & BD EC

PROVE:

AEC ABD

h) GIVEN:

D C &

DE CE

PROVE:

EDA ECB

STATEMENT REASON

STATEMENT REASON

i) GIVEN:

1 2 & BA BC

PROVE:

ABD CBD

j) GIVEN:

1 2 & EB EC

PROVE:

AEB DEC

STATEMENT REASON

STATEMENT REASON

k) GIVEN:

BD bisects ADC

BD bisects ABC

PROVE:

DAB DCB

L) GIVEN:

D C & CA DA

PROVE:

EAD BAC

STATEMENT REASON

STATEMENT REASON

2

1

C

E

A D

B

C

BE

A

D

21

D

A C

B

21

E

C

A D

B

B

D

A C

A

B

E D

C

G.CO.B.8 WORKSHEET #3 – geometrycommoncore.com NAME: ______________________ 1

1. Prove the following relationships. (involving parallel lines)

a) GIVEN:

||PT SR & TQ SQ

PROVE:

PQT RQS

b) GIVEN:

||PT SR & PT RS

PROVE:

PQT RQS

STATEMENT REASON

STATEMENT REASON

c) GIVEN:

||CB ED & A is the

midpoint of CD

PROVE:

EAD BAC

d) GIVEN:

||AB CD & ||BC DE &

C is the midpoint of AE

PROVE:

ABC CDE

STATEMENT REASON

STATEMENT REASON

e) GIVEN:

||AB CD & ||BC DA

PROVE:

ABC CDA

l) GIVEN:

||DE CB , ||DG CA &

EG BA

PROVE:

EDG BCA

STATEMENT REASON

STATEMENT REASON

Q

S

T

P

R

Q

S

T

P

R

A

B

E

DC

D

C

A E

B

4

32

1

DA

BC

G

D

E

B

C

A


What can we conclude if we know two triangles are congruent?

What does CPCTC mean?

How do the ‘PROVE’ statements differ from the previous worksheet? What do you think we have to do to

prove these sides and angles to be congruent?

Prove the following relationships.

1. GIVEN:

||AB DE &

BC DC

PROVE:

AC EC

2. GIVEN:

AC EC &

BC DC

PROVE:

B D

STATEMENT REASON

STATEMENT REASON

3. GIVEN:

AC bisects DAB & AB AD

PROVE:

B D

4. GIVEN: C is the midpoint of

AE & BD

PROVE:

AB DE

STATEMENT REASON

STATEMENT REASON

D

CA

E

B

B

C

A

E

D

D

A C

B

D

CA

E

B

G.CO.B.8 WORKSHEET #4 – geometrycommoncore.com NAME: ______________________ 1

1. Prove the following relationships.

a) GIVEN:

BC DC & AC EC

PROVE:

A E

b) GIVEN:

D C & DE CE

PROVE:

AD BC

STATEMENT REASON

STATEMENT REASON

c) GIVEN:

T is the midpoint of ME

& G J

PROVE:

T is the midpoint of GJ

d) GIVEN:

D C & CA DA

PROVE:

EA BA

STATEMENT REASON

STATEMENT REASON

e) GIVEN:

AB CB

BD bisects ABC

PROVE:

BD bisects ADC

f) GIVEN:

||CB ED &CA DA

PROVE:

BA EA

STATEMENT REASON

STATEMENT REASON

C

A E

D

B

C

BE

A

D

T

M

E

J

GA

B

E D

C

B

D

A C

A

B

E

DC

G.CO.C.10 GUIDED PRACTICE WS #1 – geometrycommoncore.com 1

Prove that the interior angles of a triangle sum to 180.

Informal Proof - Triangle Dissection

Cut out a triangle – Cut or rip each corner off – Place the three corners together (sharing the same vertex)

What do you notice?

Classic Proof – Use of Parallel Line Angle Relationships

A formal two column proof of this theorem is done using the angle relationships found with parallel lines and a

transversal. Since these angle relationships were established in objective G.CO.9, we are now able to apply

them.

Given: ABC

Prove: m1 + m2 + m3 = 180

Construct an auxiliary line parallel to AC through B.

STATEMENT REASON

AC || BD Given (Auxiliary Line)

Transformational Proof

3

2

1A C

B

D

3

2

1A C

B



Determine the measure of the angle.

a) mC = ________ b) mB = ________ c) mA = ________ d) mA = ________

Prove that Base Angles of an Isosceles are Equal (Isosceles Triangle Theorem)

Informal Proof - Paper Folding

Create an isosceles triangle by using your compass to construct a circle. Then draw two radii

(all radii of the same circle are congruent) and connect the endpoints with a segment. Cut out

the isosceles triangle.

How could you show that the base angles of the isosceles triangle are equal?

Classic Proof – Triangle Congruence

Given: ABC is an isosceles triangle, with base AC .

Prove: A C

Construct an auxiliary line that is a perpendicular bisector of AC .

(B is on the perpendicular bisector of AC because AB BC )

STATEMENT REASON

ABC is an Isosceles. Given

BD is the bisector of B Given (Auxiliary Line)

96°

44°

A

B

C

60°61°A

B

C

32°A

B

C

o

o

50°A

B

C




Prove: A C

Construct an auxiliary line that is a perpendicular bisector of AC .

Symmetry Proof


Prove: A C


Determine the measure of the requested angle.

a) mC = ________ b) mA = ________ c) mC = ________ d) mD = ________

D

B

A C

D

B

A C

98°

A

B

C

78°

A

B

C

65°

A

B

C36°

A

BC

D


Prove that the external angle of a triangle is equal to the sum of the two internal

remote angles. (Exterior Angle Theorem)

What is an external angle?

What are remote angles?

Informal Proof – Angle Dissection

On a piece of paper draw a triangle with an external angle. Cut out the triangle and external angle. Rip off the

two corners (the remote angles) that are not a linear pair to the external angle. Place these two angles at the

vertex of the external angle. What do you notice?

Classic Proof – Triangle Congruence

Given: ABC with external angle, ACD.

Prove: mACD = mB + mA

STATEMENT REASON

BC

A

D




Determine the missing information.

a) mBCD = ___________ b) mB = _________

mBCA = _________

c) mA = _________

mBCA = _________

Prove the Midsegment Theorem (that the segment joining midpoints of two

sides of a triangle is parallel to the third side and half the length)

Informal Proof

BC

A

D

74°

73°

A

B

CD

65°21°

A

B

CD

160°42°

A

B

CD

3

2

1A C

B


Transformational Proof – Translation

NYTS (Now You Try Some) Determine the missing information. a) x = ____________ b) x = ____________ c) x = ____________

ED

A

B

C

L

ED

A

B

C

x

13.5 cm3x

42 cmx

68°

G.CO.C.10 WORKSHEET #1 – geometrycommoncore.com NAME: ________________________ 1

1. Prove that the sum of the measures of the internal angles of a triangle is 180.

Given: ABC Prove: m1 + m2 + m3 = 180

Construct an auxiliary line parallel to AC through B.

2. Prove that the base angles of an isosceles triangle are equal using triangle congruence.

Given: ABC is an isosceles triangle, with base AC . Prove: A C

Construct an auxiliary line that is a perpendicular bisector of

AC .

(B is on the perpendicular

bisector of AC because AB

BC )

STATEMENT REASON

ABC is an Isosceles. Given

BD is the bisector of B Given (Auxiliary Line)

3

2

1A C

B

D

D

B

A C

G.CO.C.10 WORKSHEET #1 – geometrycommoncore.com 2

3. Prove that the exterior angle is equal to the sum of the triangle’s two remote angles.

Given: ABC with external angle, ACD.

Prove: mACD = mB + mA

STATEMENT REASON

4. The essential structure of the proof in question #1 is that the straight angle guarantees an angle sum of

180. Use a single or a sequence of transformations to also create a straight angle that might help us

establish that the sum of the internal angles of a triangle is 180 using a transformational approach. Draw in the transformation(s) that you would use and show where the straight was formed.

5. Using a transformational approach, describe how you would show that

A is congruent to C in the isosceles ABC.

6. Is it possible to perform some transformations to demonstrate that the two remote angles of the triangle are equal to the exterior angle? Explain and draw how you would do this.

3

2

1A C

B

D

B

A C

BC

A

D

BC

A

D

G.CO.C.10 WORKSHEET #1a – geometrycommoncore.com NAME: ________________________ 1

G.CO.C.10 WORKSHEET #1a – geometrycommoncore.com 2

1. DRAFT #1 -- Prove that the sum of the measures of the internal angles of a triangle is 180 using a transformational approach.

Student Commentary / Feedback

2. DRAFT #2 -- Prove that the sum of the measures of the internal angles of a triangle is 180 using a transformational approach.


3

2

1A C

B

3

2

1A C

B


3. FINAL PROOF -- Prove that the sum of the measures of the internal angles of a triangle is 180 using a transformational approach.

Teacher Commentary / Feedback

3

2

1A C

B


4. DRAFT #1 -- Prove that the base angles of an isosceles triangle are equal using transformations. ABC is an isosceles triangle where AB = CB.


5. DRAFT #2 -- Prove that the base angles of an isosceles triangle are equal using transformations. ABC is an isosceles triangle where AB = CB.


D

B

A C

D

B

A C


6. FINAL PROOF -- Prove that the base angles of an isosceles triangle are equal using transformations.

ABC is an isosceles triangle where AB = CB.


D

B

A C


7. DRAFT #1 -- Prove that the exterior angle is equal to the sum of the triangles two remote angles using transformations.


8. DRAFT #2 -- Prove that the exterior angle is equal to the sum of the triangles two remote angles using transformations.


BC

A

D

BC

A

D


9. FINAL PROOF -- Prove that the exterior angle is equal to the sum of the triangles two remote angles using transformations.


BC

A

D


1. Determine the measure of the angle.

a) mA = ________ b) mC = ________ c) mC = ________ d) mA = ________

2. Determine the measure of the angles.

a) m1 = _______ m2 = ______ b) m1 = _______ m2 = ______

m3 = _______

c) m1 = _______ m2 = ______

m3 = _______


a) x = _______ mA = _________ b) x = _______ mB = _________ c) x = _______ mA = _________

4. An exterior angle is formed between a side and the extension of a side. It will always be a linear pair with

an internal angle. In the diagram below, 4 is the exterior angle. The exterior angle theorem states that the

EXTERNAL ANGLE IS EQUAL TO THE SUM OF THE TWO REMOTE ANGLES. The remote angles are those

interior angles that are not adjacent to the exterior angle so in this case 1 & 2 are the remote angles.

m1 + m2 = m4, Explain why this would be true.

41°

99°

A

B

C

60°

65°

A

B

C

48°

A

B

C

118°

A

B

C

21

27°76° 21 3

64°

3

21

68°

2x3x - 3

68°

A

B

C

2x3x

4x

A

B

C3x + 5

2x

A

B

C

43

1

2

A

B

C



a) m1 = ______ m3 = ______ b) m1 = ______ m3 = ______ c) m1 = ______ m3 = ______

6. Find the value of x

a) x = ____________ b) x = ____________ c) x = ____________

d) x = ____________ e) x = ____________ f) x = ____________

g) x = ____________ h) x = ____________ i) x = ____________

124°3

1

49°

A

B

C

40°

3 118°

A

B

C

3 165°

98°

1

x + 5

2x - 2

xx

x

x + 40 2x - 5

3x - 17

81°

47°x

x

42°

x 82°

48°64°

AD

E

CF

B

x52°

x

121°

x

82°

59°

65°

56°

G.CO.C.10 STUDENT GUIDED PRACTICE WS #3 – geometrycommoncore.com 1

The mid-segment of a triangle is the segment

that connects the midpoints of two sides of a triangle.

THE MID-SEGMENT THEOREM

If a segment joins the midpoints of two sides of a , then the segment is parallel to the third side,

and is half its length.

VISUAL CONNECTION - FOLDING

VISUAL CONNECTION – TRANSLATION

x

2x

B

C

A

G.CO.C.10 STUDENT GUIDED PRACTICE WS #3 – geometrycommoncore.com 2

We can use this theorem to solve for different values in a triangle. Here are some examples.

x = _______ cm y = _______ cm x = ______ y = _______ cm x = _______ cm y = _______ cm

x = _______ y = _______ cm x = _____ cm y = _____ x = _______ cm y = _______ cm

x

8 cm

18 cm

y

x

76°

7 cm

y

x

18 cm

8 cm

y

x24 cm

86° yy

8 cm

x

70°

y

x

12 cm

9 cm


1. Points D and E are midpoints of their respective sides. Name two facts that you would know about the triangle using the knowledge of the mid-segment theorem.


a) b) c)

x = ______ cm y = ______ cm

x = ______ cm y = ______ cm

x = ______ cm y = ______ cm

d) e) f)

x = ______ cm y = ______ cm

x = ______ cm y = _____

x = ______ y = ______ cm g) h) i)

x = _______ y = _______

x = _______ y = _______

x = _______ y = ______ cm

E

D

A

B

C

y

8 cm x

22 cm

y

7 cm

x

10 cm

y

12.5 cmx

7 cm

8.5 cm

x

y

14 cm

70°

x

y21 cm

13 cm

62°

x

y

55°

111°

x

y

42°

68°

y

x

6 cm

y

x



a) b) c)

x = ______ y = ______ cm

x = ______ cm y = ______ cm

x = ______ cm y = ______ cm

d) e) f)

x = ______ cm y = ______

x = ______ cm y = ______ cm

x = ______ y = ______ cm

4. Determine the requested information.

a) b) c)

x = ______ cm y = ______ cm x = ______ cm y = ______ cm x = ______ cm y = ______ cm

5 cm4 cm

y

x 8 cm

8.5 cm

y

x

16 cm6 cm

y

x

62°

4 cm

6 cm

y

x

4 cm

y

x

6 cm

yx

7 cm

6 cm5 cm

y

x

14 cm

18 cm

19 cm

y

x

x

6 cm

5 cm

4 cm

y


5. The following measurements were gathered to help determine the width of the pond. What is the width of the pond?

6. Determine the value of x.

a) b) c)

x = __________ x = __________ x = __________

d) e) f)

Perimeter of mid-segment is 24 cm

x = __________ x = __________ x = __________

7. Determine the missing values. w = _______ x = ________ y = ________

18 ft

8 ft7.5 ft

2x + 3 cm

34 cm

5x + 1 cm

13 cm

5x - 4 cm

2x + 1 cm

12 cmx

60°

3x - 8 cm

x cm

10 cm

16 cm

x

w

y

x

60 cm

Documents

Geometry Workbook 5