GEO9 2.3-2.6 1

MIDPOINT THEOREM (Groups) ( Use file cards)ANGLE BISECTOR THEOREM

MIDPOINT THEOREM: If M is the mid-point of AB _____________________ then AM = 1/2 AB, MB = 1/2 AB A M

Given: If ________________________________Prove: then _______________________________

Statements Reasons .

1. M is the mid-point of AB 1. Given

2. AM MB or AM = MB 2.

3. AM + MB = AB 3.

4. AM + AM = AB 4. or 2AM = AB

5. AM = 1/2 AB 5.

6. MB = 1/2 AB 6.

ANGLE - BISECTOR THEOREM: If BX is the bisector of <ABC then m<ABX = 1/2 m<ABC, m<XBC = 1/2 m<ABC

Given: BX bisects <ABCProve: m<ABX = 1/2 m<ABC,

m<XBC = 1/2 m<ABCStatements Reasons . 1. BX bisects <ABC 1.

2. 2.

3.

4.

5. 5.

6.

B

C

X

A

B

GEO9 2.3-2.6 22.3 Midpoint Theorem

Angle Bisector Theorem

Given: Prove:

1.

J

O

KM

P

Use the pieces given on next page to arrange in the correct order

GEO9 2.3-2.6 3

GIVEN

, bisects JK and MK

JK = MK

JO = PK

PK JO

MULT =

MIDPOINT THEOREM

SUBSTITUTION

DEF OF CONGRUENCE

GEO9 2.3-2.6 4

2. Given: , bisects <AFE bisects <DEF

Prove:

A

21

F E

DCBA

GEO9 2.3-2.6 5Prove with your group by filling in the blanks. 3) Given: AB = AC _Prove: BD EC

1. AB = AC 1. Given D, E are midpoints of AB, AC

.

2. AB = AC 2.

3. BD = 3. Midpoint Theorem EC =

4. BD = EC 4.

5. 5.

Given: <ABC = <ACB, BE bisects <ABC, CD bisects <ACBProve: <2 <4

1. <ABC = <ACB 1.Given BE bisects <ABC CD bisects <ACB

2. <ABC = <ACB 2.

3. <2 = 3. <4 =

4. 4. Substitution

5. 5.

A

B C

D E

F1

2

3

4

A

B C

D E

F1

2

3

4

GEO9 2.3-2.6 62-4 SKETCHPAD

Theorum 2-3: Vertical Angles are Congruent.

Given: Ð1 and Ð3 are vert Ð's. Prove Ð1 Ð3

Statements Reasons1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

I Given mÐT, find its supplement and complement.

1. mÐT=40° supp = _______ comp = ________2. mÐT=1° supp = _______ comp = ________3. mÐT=4x supp = _______ comp = ________

II Complete with always, sometimes or never.

1. Vertical angles __________ have a common vertex.2. 2 right angles are __________ complementary.3. Right angles are __________ vertical angles.4. Vertical angles __________ have a common supplement.

III In the following drawing, if you were given Ð2 Ð3, how could you prove Ð1 Ð4?

DISCUSS WITH YOUR NEIGHBOR OR GROUP

1) In the diagram, . Name two other angles1 2 3

45

6

GEO9 2.3-2.6 7congruent to .

1) Given

2) Vertical Angle Theorem(VAT) 3) Transitive

4) VAT

5) Transitive

2) Given: Prove:

1.) 1.) Vertical Angle Theorem

2) 2) Given

3.) 3.) Vertical Angle Theorem

4.) 4.)

3) If ,

4) Find the measure of an angle that is 5) twice as large as its supplement.

2x+5 4x-35

Find x

6)1.) 2.)

1

2

4

3

AB

C

DE

F

O

GEO9 2.3-2.6 8

3.) 4.)

5.) 6.)

7.) 8.) 9.) 10.)

A

GEO9 2.3-2.6 92-5 Perpendicular Lines

Definition If two lines intersect to form __________________or____________________,then _____________________________________________________________________SKETCHPAD

Theorum 2-4 If two lines are perpendicular, then

Theorum 2-5 If two lines form congruent adjacent angles, then_____________________________________________________________

Theorum 2-6 If the exterior sides of two adjacent acute angles are perpendicular, then

SKETCHPAD

(for problems 1-3)

1. If , then: ____________________________ or _________________________

Why? ____________________________ _______________________

2. If Ð1 is a right angle, then: _____________________ or ________________________ Why? ____________________________________ _________________________

3. If Ð1 Ð2 then: ______________________________ or ________________________ Why? _____________________________________ _________________________

4. If, then: ___________________________ or __________________________ Why? _____________________________________ _________________________

1 2

A

B CD

1 2

A

B CO

D

GEO9 2.3-2.6 10

5. If, then:

, are the following statements true or false?

1. .2. ÐCGB is a right angle.3. ÐCGA is a right angle.4. mÐDGB = 90°5. ÐEGC and ÐEGA are complements.6. ÐDGF is complementary to ÐDGA.7. ÐEGA is complementary to ÐDGF

8) , , m< BOC = 36 .

a) Name some complementary angles.

b) Name some congruent angles.

c) Find all the angles.

A

B

C

F

DE

H

G O

GEO9 2.3-2.6 11

2-6 Planning Proofs

Theorum 2-7 If two angles are supplements of congruent angles, (or of the same angle) then

Given: Ð1 and Ð2 are supp Ð3 and Ð4 are supp Ð2 Ð4Prove: Ð1 Ð3

Statements Reasons

1) Ð1 and Ð2 are supp 1) Given Ð3 and Ð4 are supp Ð2 Ð4

Theorum 2-8 If two angles are complements of congruent angles, ( or the same angle ), then

Which angles are congruent?

12

34WARNING!

41

23

AB

C

D

EF

G

GEO9 2.3-2.6 122-4 to 2-6

1. Given: ÐO is comp to Ð2 ÐJ is comp to Ð1 Prove: ÐO ÐJ

Statement Reason

2. Given: Ð1 Ð3 Prove: Ð2 is supp to Ð3

Statement Reason

3. Given: Ð1 is comp to Ð3 Ð2 is comp to Ð4 Prove: Ð1 Ð4

Statement Reason

O

K

J

1M

H

2

12

3

4

123

LOOK FOR PATTERNS!!!

GEO9 2.3-2.6 13

4. Given: Ð1 Ð4 Prove: Ð2 Ð3

Statement Reason

5. Given: ÐA is comp to ÐC ÐDBC is comp to ÐC Prove: ÐA ÐDBC

Statement Reason

D

C

A

B

1 2 3 4

GEO9 2.3-2.6 14

More Chap 2 Proofs

1. Given: AB BC DC BC , , Ð1 Ð4 Prove: Ð2 Ð3

2. Given: Ð3 Ð4, Ð1 Ð2, Ð5 Ð6 Prove: Ð1 Ð6

3. Given: Ð1 Ð6 Prove: Ð5 Ð2

4. Given: Ð1 is supp to Ð5 Prove: Ð3 Ð2

5. Given: Ð3 Ð2, AB AE DE AE , Prove: Ð1 Ð4

6. Given: ACbisects ÐBCD, Ð3 is comp to Ð1, Ð4 is comp to Ð2 Prove: Ð1 Ð2

C

D

41

1 432 5

4

5

6

A

B32

12

3

36

3 5

4

4

21

5

21

D

C

EA

B

6

4

D

3

2A

1

B

C

GEO9 2.3-2.6 15GEOMETRY REVIEW

CH 2.2- 2.6

(1) Supply a reason to justify each statement in the following sequence if

(a) Ð 1 Ð BFD(b) Ð 2 and Ð 3 are complementary (c) mÐ 2 + mÐ 3 = 90(d) Ð 1 is a right angle(e) mÐ 1 = 90(f) mÐ 2 + mÐ 3 = mÐ 1(g) mÐ BFD = mÐ 2 + mÐ 3 (k) mÐ 4 + mÐ 5 = 180(h) mÐ 3 = mÐ 5 (l) Ð 4 and Ð 5 are supplementary(i) mÐ 2 + mÐ 5 = 90 (m) mÐ 1 + mÐ 2 + mÐ 3 = 180(j) Ð 2 and Ð 5 are complementary (n) AF + DF = AD

(2) Given the figure to the right, , mÐAFD = 155° , mÐ 2 = 4 mÐ 3 , find the measures of all the numbered angles.

(3) Given the figure as marked, find the values of x and y.

C

D

EF1

2

3

G

45

A

B

6

(3x + 5)°

(2x + 5)°

(3x + y)°

(6y + 15)°

C

B

A D

E

F321

54

GEO9 2.3-2.6 16

(4) Find the measure of an angle if 80° less than three times its supplement is 70° more than five times its complement.

(5) (6)

Given: Ð 1 Ð 3 Given: Ð 1 and Ð 7 are supplementary

Prove: Ð 2 Ð 4 Prove: Ð 6 Ð 3

(7) (8)

Given: bisects Ð DAB Given: bisects Ð CAE Ð 1 Ð 4

Prove: Ð 1 Ð 3 Prove: Ð 2 Ð 3

(9) (10)

Given: bisects Given: mÐ 1 = mÐ 3 bisects mÐ 2 = mÐ 4

AB = AE

Prove: BC = DE Prove: mÐ 5 = mÐ 6

1 2 34

567 8

1 2

6

3

7 8

4

5

A

B

2 3

E

1

C D

A

B E

C D

A

B

E

C

D

12

5 6

34

F

A B

3

41

2

E F

C

D

GEO9 2.3-2.6 17

(11)Refer to the figure to the right.

Given: mÐ 1 = mÐ 2 AB = BC Ð 3 is a right angle

Supply a “reason” to justify each statement made in the following “sequence”.

(1) B is the midpoint of _____________________________________________________

(2) AC = AB_______________________________________________________________

(3) _________________________________________________________

(4) mÐ ABE = mÐ 2________________________________________________________

(5) Ð 1 Ð 4________________________________________________________________ (6) mÐ 3 = 90_______________________________________________________________

(7) ________________________________________________________________ (8) mÐ 3 = mÐ ABE__________________________________________________________ (9) mÐ ABE = 90____________________________________________________________(10) mÐ 1 + mÐ 2 = mÐ ABE___________________________________________________(11) Ð 1 and Ð 2 are complements_______________________________________________(12) mÐ 1 + mÐ 2 = 90________________________________________________________(13) mÐ DBC + mÐ 4 = 180____________________________________________________(14) Ð DBC and Ð 4 are supplements_____________________________________________(15) mÐ 2 + mÐ 3 = mÐ DBC___________________________________________________(16) mÐ 2 + mÐ 3 + mÐ 4 = 180_________________________________________________(17) mÐ DBC = mÐ 5_________________________________________________________(18) mÐ 2 + mÐ 3 = mÐ 5______________________________________________________(19) Ð 4 and Ð 2 are complements_______________________________________________(20) mÐ 1 + mÐ 5 = 180_______________________________________________________ (21) mÐ 1 + mÐ 2 + mÐ 3 + mÐ 4 + mÐ 5 = 360___________________________________ (22) BD + BF = DF____________________________________________________________

A

CB

D

E

F

23

4

5

1

GEO9 2.3-2.6 18

CH 2.2-2.6

DEFINITIONS

1) Compementary -

2) Supplementary –

THEOREMS

1) VAT –

2) Midpt. Th –

3) < Bis Th –

4) If , then

5) If , then

6) If ext , then

7) SAT

8) CAT

GEO9 2.3-2.6 19SUPPLEMENTARY PROBLEMS CH 2

CH 2.3

1) The point on segment AB that is equidistant from A and B is called the midpoint of AB. For each of the following, find the coordinates for the midpoint of AB:(a) A (-1, 5 ) and B ( 5, -7 ) (b) A ( m, n ) and B ( k, r )

2) Fold down a corner of a rectangular sheet of paper. Then fold the next corner so that the edges touch as in the figure. Measure the angle formed by the fold lines. Repeat with another sheet of paper, folding the corner at a different angle. Explain why the angles formed are congruent.

3) Given triangle ABC with vertices A=( 2, 2 ) B = ( 10, 4 ) and C = (8, - 4 ). Find the midpoints of ( call it X ) and ( call it Y ). Find the distance

a) AB b) BC c) A to the midpoint X d) C to the midpoint Y e) the length of the segment connecting the midpoints X and Y.

f) XYWhat conclusions can you make , describe them using the word segments in your description.

Ch 2.4-6 SUPPLEMENTARY PROBLEMS

4) Graph the lines 2x – y = 5 and x + 2y = -10 on a piece of graph paper on the same set of axes. Use a protractor to measure the angle of intersection.

5) When two angles fit together to form a straight angle, they are called supplementary angles, and either angle is the supplement of the other When two angles fit together to form a right angle, they are called complementary angles, and either angle is the complement of the other. What is the supplement of an angle that measures x degrees? What is the complement of an angle that measures x degrees?

6) You have probably heard the statement that the three angles of a triangle add together to equal 180 degrees. Is this is true, what can be said about the two non-right angles in a right triangle? Write an argument that supports your conclusions.

12

3 4

GEO9 2.3-2.6 207) Let P = (a, b), Q = ( 0,0) and R = ( -b, a), where a and b are positive numbers. Prove that

angle PQR is right, by introducing two congruent right triangles into your diagram. Verify that the slope of segment QP is the negative reciprocal of the slope of segment QR.

8) Given the following diagram:(a) If m<DBA is 150 degrees and m<ACB is 30 degrees, find the measure of < ABC and <ACE.(b) If m<DBA is equal to m<ACE, come up with a rule to find m<ABC and m<ACB. (c) If you had two right triangles and one acute angle in each was equal, would the others have to be equal? Explain. (d) Come up with a rule to explain this.

B CD E

A