- 1 - Geometry Rules! Chapter 4 Notes

Notes #22: Section 4.1 (Congruent Triangles) and Section 4.4 (Isosceles Triangles) Congruent Figures Corresponding Sides

Corresponding Angles

***_________________ parts of_______________ triangles are _____________ *** Practice: 1.) If ∆CAT ≅ ∆DOG, then complete: (draw a picture first) m C∠ = _____ TCA∆ ≅ _____ GD ≅ _____ O∠ ≅ _____ TA = _____ ODG∆ ≅ _____ 2.) ZAK JOE∆ ≅ ∆ a) Name three pairs of corresponding angles: b) Name three pairs of corresponding sides:

- 2 - 3.) The two triangles shown are congruent; complete. (It will help to rotate the triangles first, to get them in corresponding positions) a) _____RAV∆ ≅ b) _____R∠ ≅ c) EV = _____ d) _____m A∠ = e) _____NV ≅ f) _______VRA∆ ≅

V

R A

E N

Isosceles Triangles Isosceles Triangle Theorem ( ) If two sides of a triangle are congruent, then the angles opposite them are ___________. Converse of the Isosceles Triangle Theorem ( ) If two angles of a triangle are congruent, then the __________ opposite them are _________.

- 3 - Equilateral Triangles Practice: Solve for x and y 4.)

7y - 53y + 7x

40

5.)

y

x

6.)

64

58

9

12y

x

7.)

5x - 10

3x - 2y

100

40

8.) In equilateral ∆XYZ, m X a b∠ = + and 2m Y a b∠ = − . Find a and b.

9.) In equiangular ∆ABC, AB = 2x + y, BC = 6x – 2y, and AC = 10. Solve for x and y.

10.) What can you conclude from the picture?

- 4 -

10 cm10 cm10 cm10 cm

E

F

G C B

A

11.) Given: is the midpoint of C BD 1 2∠ ≅ ∠ Prove: AB CD≅

Statements

21A

B

D

C

Reasons

1.) 2.)

3.) AB BC≅ 4.)

1.) 2.) Definition of Midpoint 3.) 4.)

12.) Given: 1 4∠ ≅ ∠ Prove: AB BC≅

Statements

432

1CA

B

Reasons

1.) 2.)

1.) 2.)

- 5 - 3.) 4.)

3.) Substitution 4.)

- 6 - Notes #23: Sections 4.2 and 4.5 (Methods of Proving Triangles Congruent) Q: How can we prove that two triangles are congruent to each other? A: Five ways: SSS, SAS, ASA, AAS, HL SSS: _______-________-________ Postulate

SAS: _______-________-________ Postulate

ASA: _______-________-________ Postulate

AAS: _______-________-________ Postulate

HL: _____________-______________-(_______________) Postulate

- 7 - Are the triangles congruent? If so, write the congruence and name the postulate used.

• Redraw your triangles so they line up • You need three congruent pairs of sides/angles to follow:

SSS, SAS, ASA, AAS, or HL

• Look for “hidden” pieces in: - vertical angles - overlapping sides

- congruent angles formed by parallel lines - bisected angles - ITT/Converse of ITT

- midpoints 1.)

P

O

QR

T S

______ ______ by ________∆ ≅ ∆

2.) V

U WX

______ ______ by ________∆ ≅ ∆

3.)

80

80

5 in

5 in 7 in

7 in

Y

Z

X

B A

C

______ ______ by ________∆ ≅ ∆

4.)

S

R T

B

A

C

______ ______ by ________∆ ≅ ∆

5.)

F

E

D

G

H

______ ______ by ________∆ ≅ ∆

6.)

F

E

D

G

H is the midpoint of

and

F

DG EH

______ ______ by ________∆ ≅ ∆

- 8 - 7.)

V

U WX

______ ______ by ________∆ ≅ ∆

8.)

M

A

T

H bisects MT AMH∠

and ATH∠

______ ______ by ________∆ ≅ ∆ 9.)

D

A

B

C ______ ______ by ________∆ ≅ ∆

10.)

V

U WX

______ ______ by ________∆ ≅ ∆ 11.) Given: , WX YZ XY ZW≅ ≅ Prove: WXY YZW∆ ≅ ∆

Statements

X

W

Y

Z

Reasons

1.) 2.) 3.)

1.) 2.) 3.)

- 9 - 12.) Given: , WX YZ WX YZ≅ Prove: WXY YZW∆ ≅ ∆

Statements

X

W

Y

Z

Reasons

1.) 2.) 3.) XWY ZYW∠ ≅ ∠ 4.)

1.) 2.) Reflexive 3.) 4.)

Notes #24: More Proofs and Section 4.3 (Using Congruent Triangles) Are the triangles congruent? If so, write the congruence and name the postulate used. 1.)

X

W

Y

Z

, WX YZ XY ZW≅ ≅

2.) X

W

Y

Z, WX YZ WX YZ≅

3.)

X

W

Y

Z, WX YZ XY ZW

4.) X

W

Y

Z

, WX YZ XY ZW≅

- 10 - 5.) Complete: a) ∆ABC ≅ __________ because _______ b) AB = ____ because ___________ c) AC = EC because ___________. Then C is the midpoint of _________ by _______________________________.

CA E

B

D

d) A∠ ≅ _____ because _________. Then AB ED because _______________________. Complete the proofs: follow these key steps

1. Re-draw and label your picture; mark congruencies 2. Find and list 3 congruencies: shared sides (reflexive) vertical angles alternate interior/corresponding angles (only when lines are ) angle bisectors midpoints ITT 3. State ∆≅∆ by SSS, SAS, ASA, AAS, or HL 4. State part ≅ part by CPCTC

6.) Given: , WX YZ XY ZW≅ ≅ Prove: X Z∠ ≅∠

Statements

X

W

Y

Z

Reasons 1.) 2.) 3.) _______ _______∆ ≅ ∆

1.) 2.) 3.)

- 11 - 4.) 4.) 7.) Given: , WX YZ YX WZ Prove: XY ZW≅

Statements

X

W

Y

Z

Reasons 1.) 2.) 3.) 4.) _______ _______∆ ≅ ∆ 5.)

1.) 2.) If two parallel lines are cut by a transversal, then ____________________ angles are congruent. 3.) 4.) 5.)

8.) Given: is the midpoint of and C AD BE Prove: A D∠ ≅ ∠

Statements

C

B

A

D

E

Reasons

1.) 2.) 3.) 4.) _______ _______∆ ≅ ∆

1.) 2.) Definition of Midpoint 3.) 4.)

- 12 - 5.)

5.)

9.) Given: bisects and CT ACS ATS∠ ∠ Prove: A S∠ ≅ ∠

Statements

C T

A

S

Reasons 1.) 2.) 3.) 4.) 5.)

1.) 2.) Definition of __________ ___________ 3.) 4.) 5.)

10.) Given: 1 2, is the midpoint of X WY∠ ≅ ∠ Prove: WX YZ≅

Statements

21X Z

Y

W

Reasons 1.) 2.) 3.) 4.)

1.) 2.) Definition of ______________ 3.) 4.)

- 13 -

Notes #25: Algebra and Proof Review: 1.) Graph the points and name the quadrant in which each point is found: A(-3, 2) B(0, -7) C(4, -1) D(6, 0) 2.) Evaluate for a = -2 and b = 3

-a2 – b(ab – 3)

3.) Simplify:

-32 – 4(22 – 1) – (-3)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10

-9-8

-7

-6-5

-4

-3-2

-1

12

3

45

6

7

89

10y

4.) In equilateral ABC∆ , 2 4m A x y∠ = + and

5m B x y∠ = + . Solve for x and y.

5.) Solve for x and y

80

50

8

12y

x

6.) Solve for x and y

7x - 3

3x + 5y

120

30

7.) What does CPCTC stand for? 8.) KIM BEN∆ ≅ ∆ Complete:

a) _____IK = b) _____I∠ ≅ c) _____ENB∆ ≅

- 14 - d) _____IK =

Complete each proof by filling in the blanks.

1. Given: AB || DE AB ≅ DE

Prove: ∆ABC ≅ ∆EDC

1. 1. Given

2. 2. ll lines cut by a

_____________ trans form ≅ alt. int.

∠’s

3. 3.

2. Given: AB ≅ CD AB || CD

Prove: ∆ADB ≅ ∆CBD

1. 1.

2. 2. Reflexive

3. 3. ll lines cut by a

trans form ≅ alt. int.

∠’s

4. 4.

3. Given: E is the mdpt of TP and MR

Prove: TM ≅ PR

1. 1. Given

2. TE ≅ PE 2.

____________

3. 3.

4. ∆ ≅ ∆ 4.

5. 5.

4. Given: ∠1 ≅ ∠4; ∠2 ≅ ∠3 M is the mdpt. of AB

Prove: AC ≅ BD

1. 1.

A B

C

D E

A

B

D

C

T

M

E R

P

A

C D

B M 1 2 3 4

15 2. AM ≅ BM 3.

3. ∆ ≅ ∆ 3.

4. 4.

5. Given: AD || ME MD || BE M is the mdpt.

of AB

Prove: MD ≅ BE

1. 1.

2. ∠2 ≅ ∠4 2.

3. 3.

4. ∆ ≅ ∆ 4.

5. 5.

6. Given: WO ≅ ZO XO ≅ YO

Prove: ∠W ≅ ∠Z

1. 1.

2. 2.

3. ∆ ≅ ∆ 3.

4. 4.

7. Given: RS ≅ RT

Prove: ∠3 ≅ ∠4

1. 1. Given

2. 3. ITT

3. ∠3 ≅ ∠1 2. ∠4 ≅ ∠2

4. 4.

8. Given: M is the mdpt of JK ∠1 ≅ ∠2

Prove: JG ≅ MK

1. 1.

2. KM ≅ JM 2.

A

M

B

D

E

1

2

3

4

W

X Y

Z

O

R

T S

4

2 1

3

1

2

G

M

J

K

16 3. JM ≅ JG 3.

4. 4.

17 Notes #26: Section 4.7 (Special Segments in Triangles) and Proof Review

Median: connects a ___________ to the __________ of the opposite side

A

B

C

A

B

C

________ and _________ are medians of __________ _____is the _____________ of AC _____ is the _____________ of BC ______ = ______ ______ ≅ ______ ____ is equidistant from ___ and ___ ____ is equidistant from ___ and ___

Altitude: a ________________ segment from a vertex to an opposite side

A

B

C

A

B

C

________ and _______ are ______________ of ∆ABC ____ ⊥ _____ ____ ⊥ _____

_________ 90m∠ = _________ 90m∠ =

18

Perpendicular Bisector: a _______________________ segment to the ________________of the opposite side

A

B

C

A

B

C

______ and ______ are _________________ _________________ of ∆ABC _____is the _____________ of AC _____is the _____________ of BC ____ ≅ _____ ____ = _____ ____ ⊥ _____ ____ ⊥ _____ If X is on the perpendicular bisector to AC then X is equidistant from ______ and _______.

Angle Bisector: cuts a ___________ ____________ into two equal ____________ AND is ________________________ from the sides of the angles.

A

B

C

A

B

C

__________ are _____________________ of ∆ABC

_______ _________m m∠ = ∠

_______ _________∠ ≅ ∠ _____ is equidistant from _____ and ______. _____ is equidistant from ______ and ______.

19 Practice: 1.) XZ is a median to WY . 5 3WZ x= − and

22ZY = . Solve for x.

W Y

X

Z

2.) BD is a perpendicular bisector to AC . 3 15m BDA x∠ = − , 2 6AD y= + and

4 14DC y= − . Solve for x and y.

B

A CD

3.) Name an angle bisector, a median, and an altitude of ABC∆ Angle bisector: _________ Median: _________ Altitude: _________

A B

C

Z

X

Y

20

Proof Review

1.) Given: X is the mdpt. of CB

CD AB

Prove: AB ≅ DC

1. ____ 1. Given

2. _____ ___________ 2.

3. _____ 3.

4. 4.

5. 5.

2. Given: LM ≅ JK LM || JK

Prove: JM ≅ LK

1. LM ≅ JK; LM || JK 1. Given

2. 2. If parallel lines are cut by a transversal,

then alternate interior angles are congruent.

3. 3.

4. _____________ 4.

5. 5.

3.) Given: , WX ZY WY ZX≅ ≅ Prove: WX ZY

43

21

Z

W

Y

X

1.__________________________

2.__________________________

3.__________________________

4.__________________________

5.__________________________

1._________________________

2._________________________

3._________________________

4._________________________

5._________________________

J K

L M

A B

D C

X

21 4.) Given: 1 3∠ ≅ ∠ Prove: ON OP≅ 3

21 PN

O

1.__________________________

2.__________________________

3.__________________________

4.__________________________

1.__________________________

2.__________________________

3.__________________________

4.__________________________

Are the triangles congruent? If so, write the congruence and name the postulate used. 5.)

F

E

D

G

H

______ ______ by ________∆ ≅ ∆

6.)

F

E

D

G

H is the midpoint of

and

F

DG EH

______ ______ by ________∆ ≅ ∆ 7.)

V

U WX

______ ______ by ________∆ ≅ ∆

8.)

M

A

T

H bisects MT AMH∠

and ATH∠

______ ______ by ________∆ ≅ ∆ 9.)

D

A

B

C ______ ______ by ________∆ ≅ ∆

10.)

V

U WX

______ ______ by ________∆ ≅ ∆

22 Notes #27: Test and Algebra Review 1.) Solve:

( )3 1 35

x x− = +

2.) Get y alone: 2 5 8x y+ =

3.) Add:

( ) ( )2 26 4 3 2 3 1x x x x+ − + − +

4.) Subtract:

( ) ( )2 26 4 3 2 3 1x x x x+ − − − +

5.) Simplify:

2 22 (4 ) 5 ( 7 )x x x x x x+ − −

6.) Distribute:

-2xy(4x2 + 6y - 3)

7.) Evaluate: if a = -2 and b = 4

-2a2 – 3ab(a + 2b)

8.) Evaluate: if x = -1 and y = 3

-y2 – xy(y – x)

9.) FOIL:

( 3)( 4)x x− +

10.) FOIL: (3 1)(2 4)x x− +

11.) FOIL: (5 2)(2 1)x x− +

12.) FOIL: 3 ( 4)( 1)x x x− +

23 Chapter 4 Study Guide: 1. Given: ,WX ZY XY WZ Prove: X Z∠ ≅ ∠

43

21 Z

YX

W

Statements Reasons

1. _________________________ 1. ________________________ 2. _________________________ 2. ________________________ _________________________ ________________________ 3. _________________________ 3. ________________________ 4. _________________________ 4. ________________________ 5. _________________________ 5. ________________________ 2. Given: ,WX ZY XY WZ≅ ≅ Prove: XY WZ

43

21 Z

YX

W

Statements Reasons

1. _________________________ 1. ________________________ 2. _________________________ 2. ________________________ 3. _________________________ 3. ________________________ 4. _________________________ 4. ______ CPCTC__________ 5. _________________________ 5. ________________________ 3. Given: ,C is the midpoint of BEAB DE Prove: AC CD≅ E

D

C

B

A

Statements Reasons

1. _________________________ 1. ________________________ 2. _________________________ 2. ________________________ 3. _________________________ 3. ________________________ 4. _________________________ 4. ________________________ 5. _________________________ 5. ________________________ 6. _________________________ 6. ________________________ 4. Given: ,NO PO MO QO≅ ≅ Prove: M Q∠ ≅ ∠

Q

P

O

N

M

Statements Reasons

1. _________________________ 1. ________________________ 2. _________________________ 2. ________________________ 3. _________________________ 3. ________________________ 4. _________________________ 4. ________________________

24

FE

D

C

B

A

5. Name each of the special segments: (a) median (b) altitude (c) angle bisector 6. Complete: (a) F is equidistant from _____ and _____. Therefore, _____ ≅ ________

(b) Any point on AD is equidistant from ______ and _______.

7. ABC is equilateral. If 2m A x y∠ = + and 4m B x y∠ = − , solve for x and y. 8. In ,XYZ XY YZ≅ . If 5 10 and 2 44m X x m Z x∠ = − ∠ = + solve for m X∠ 9. Are the pairs of triangles congruent? If so, name the congruence and the postulate used. a) b)

c) d)

10. a) Solve for x: b) Solve for y:

30 2x + 17

6x - 76458

C

B

A

2y + 8

3y - 6

12

C

B

A

25 11.) Solve each system… a) Using Elimination x + 2y = 2 3x – 3y = -12

b) Using Substitution x – 3y = -4 2x + 2y = 4

12.) The sum of twice an angle’s complement and it supplement is 249 degrees. Find the angle, its complement, and its supplement.

13.) The difference of twice a number and eight is four more than three times the number. Find the number.

14.) QX bisects PQR∠ , 4 11m PQX x∠ = − , 2 5m XQR x∠ = + . What kind of angle is PQR∠ ?

15.) The measure of one angle of a triangle is ten more than twice the smaller angle. The third angle of the triangle is ten less than six times the smallest angle. Find the measure of all three angles and then classify the triangle. (Hint: let x = the smallest angle)

16.) Evaluate a) (p – px) + (a + p) for a = -3, p = 2, x = -4

b) -32 – 2[2(5 – 3) – (2 – 3)]

17.) If AH is the perpendicular bisector of MT , solve for x and y. 4 18m MHA x∠ = +

5 34 15

MH yHT y

= += +

M

A

TH

26