MIDPOINT THEOREM (Groups) ( Use File Cards)

GEO9 2.3-2.6 20

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 B

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 <ABC

Prove: m<ABX = 1/2 m<ABC,

m<XBC = 1/2 m<ABC

Statements Reasons .

1. BX bisects <ABC 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

2.3 Midpoint Theorem

Angle Bisector Theorem

Given: Prove:

2. Given: , bisects <AFE

bisects <DEF

Prove:

Prove 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 <ACB

Prove: <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.

2-4 SKETCHPAD

Theorum 2-3: Vertical Angles are Congruent.

Given: Ð1 and Ð3 are vert Ð's.

Prove Ð1 @ Ð3

Statements / Reasons
1. / 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 angles

congruent 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

1.) 2.)

3.) 4.)

5.) 6.)

7.) 8.)

9.) 10.)

2-5 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? ______

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.

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 @ Ð4

Prove: 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?

2-4 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

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

More Chap 2 Proofs

1. Given: , 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,

Prove: Ð1 @ Ð4

6. Given: bisects ÐBCD,

Ð3 is comp to Ð1, Ð4 is comp to Ð2

Prove: Ð1 @ Ð2

GEOMETRY 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

(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.

(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

(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______

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

SUPPLEMENTARY 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) XY

What 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.

7) 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.

would the others have to be equal? Explain.

(d) Come up with a rule to explain this.