BeginningSegment

ProofsName: _________________

Period: ______

Teacher: _______________

Table of Contents

Day 1 : SWBAT: draw conclusions by applying geometric definitions. Pages 1-4 HW: page 5

Day 2: SWBAT: draw conclusions by applying geometric definitions. Pages 6-7 HW: pages 8-9

Day 3: SWBAT: apply definitions and theorems to write geometric proofs. Pages 10-11 HW: pages 12-13

Day 4: SWBAT: apply the properties of equality and the Addition Postulate to write geometric proofs.

Pages 14-19 HW: pages 20-22

Day 5: SWBAT: apply the Subtraction postulate to write geometric proofs. Pages 23-26 HW: pages 27-29

Day 6: SWBAT: apply the multiplication and Division postulate to write geometric proofs.

Pages 30-33 HW: pages 34-37

Day 7: SWBAT: apply postulates, definitions and theorems to write geometric proofs.

Pages 38-43

Day 8: SWBAT: apply postulates, definitions and theorems to write geometric proofs.

Pages 44-45

Day 9: Exam

The Building Blocks of ProofAll from www.wikipedia.com

Part of the problem you may have with this unit is that many of the things I will ask you to formally prove is either (a) intuitively obvious, or (b) has been proven by someone else already. But that is the burden of proof, to prove even the painfully obvious, using only the smallest accepted truths, until you have shown without a shadow of a doubt that you’ve attained an irrefutable truth.

Definitions:A definition is a statement of the meaning of a word. There is a big difference between the word “definition” and “property.” For example, there may be one acceptable definition for the word “parallelogram,” but it has many properties.

Postulate (or Axiom):“a statement or assumption that is agreed by everyone to be so obvious or self-evident that no proof is necessary, and which can be used to prove other statements or theorems.”

Example: Partition postulate – A whole is equal to the sum of its parts.

Theorem:“a proposition that has been… proved on the basis of explicit assumptions.” A key property of theorems is that they possess proofs, not merely that they are true.”

Example: Pythagorean theorem – the sum of the squares of the legs of a right triangle are equal to the square of its hypotenuse.

Corollary:“a proposition that follows with little or no proof from one other theorem or definition. “

ExamplesTheorem: If 2 sides of a triangle are congruent, then the angles opposite those sides are also congruent.Corollary: If a triangle is an equilateral triangle, then it is equiangular.

1

Review of Definitions

1: The midpoint of a line segment divides the segment into two congruent segments.

Ex: If M is the midpoint ofAB , then _____________________________________________.

2: The bisector of a line segment intersects a line segment at its midpoint.

Ex: If D⃗E bisectsAB , then _____________________________________________.

3: An angle bisector divides an angle into two congruent angles.

Ex: If B⃗E bisects ABC, then ___________________________________________.

2

4. Perpendicular Lines (segments) intersect to form right angles.

Ex: If AB⊥CD then _____________________________________________.

** Note **

5. The perpendicular bisector of a line segment is perpendicular to the line segment and bisects the line segment.

Ex: Then __________________ and ______________________.

6. An altitude is a line segment drawn from any vertex of the triangle that is perpendicular to and ending in the line containing the opposite side.

Ex: , then ________________________

3

7. A median is a line segment drawn from any vertex of a triangle to the midpoint of the opposite side.

Ex: Then ______________________.

8.

Ex: If ∆ABC is an equilateral triangle, then _________________ and ____________________.

9.

Ex: If Then _________________________________.

Ex: If Then _________________________________.

4

5

6

∡3 are supplementary

Homework – Day 1

7

Day 2

8

Day 2 - HW

1. Write a conclusion using ¿ .

a) If B is the midpoint of AD , then _____________.

b) If B⃗O bisects AD , then _____________.

c) If B⃗Ebisects ∡DBO, then _____________.

d) If B⃗Ebisects AD , then _____________.

2. Given: HE is the angle bisector of CHF

F is the midpoint of CG

a) Name two congruent segments _____________

b) Name two congruent angles _____________

c) CE + EF = _____________.

d) FG + EF = _____________.

e) CF - EF = _____________.

f) m∡CHF - m∡EHF = _____________.

g) ∡CHE + ∡EHG ¿ _____________.

3. State whether each of the following statements is True or False.

Given: ∡APQ is a right angle. P is the midpoint of AB

a) RPbisects AB

b) ∡1 is a right angle.

c) AB + AP = PB

d) AP=PB

e) QPbisects AB

f) A line has exactly one bisector.

g) If RP = PB then P is the midpoint of RB

h) The endpoint of AB is A.

i) ∡APR and ∡RPB is equal to 180.

j) ∡1 is adjacent to ∡3.

9

H

E FGC

k) ∡1 and ∡2 form a linear pair.

4. Write a valid conclusion(s) based on the given information.

a) DC bisects ∡ACB _____________________

b) BE is an altitude to ∆ACB _____________________

c) DC and BE bisect each other__________________

5. Write a valid conclusion based on the given information.

a) F is the midpoint of DC _________________________

b) AE is the perpendicular bisector of FD _________________________________

c) CF + FE = _____________

d) AB – AF = _____________

3. Write a valid conclusion based on the given information.

a) SF is a median to ∆SAD ____________________

b) DLbisects SF ____________________

c) DL is an angle bisector____________________

4. Write a valid conclusion based on the given information.

a) TM is the perpendicular bisector of GP____________________

b) ∡TMP - ∡TMR ¿ ____________________

c) ∡TMQ + ∡QMG ¿ ____________________

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11

Day 3 - Proofs with Definitions

1. Given: RT bisects QS at M.

Prove: QM≃MS

2. Given: JT⊥CP Prove: ∡CJT ¿∡PJT

3. Given: D⃗B bisects ∡ABCProve: ∡ABD ¿∡DBC

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____ ________ is the midpoint of ____

______ _______

____ and ____ are right ∡s

____ ____

4. Given: OR is a median to ∆OPS

Prove: PR¿ RS

5. Given: AM is an altitude to ∆ABCProve: ∡BMA¿∡AMC

13

O

PS

R

A

B CM

____ ________ is the midpoint of ____

____ and ____ are right ∡s

____ ____

____ ____

Day 3 - Homework

1. 2.

3. 4.

5. 6.

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(3) (3)

∡1∧∡2are ¿∡s

(3) (3)

(3) (3)

(3) (3)

(4) (4)

7.

8.

9.

10.

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Day 4 - Proving Segment Relationships

A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.

You learned in Chapter 1 that segments with equal lengths are congruent and that angles with equal measures are congruent. So the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. An important part of writing a proof is giving justifications to show that every step is valid.

Example 1:

16

Example 2:

Example 3:

17

If equal quantities are added to equal quantities, the sums are equal.

ORIf congruent quantities are added to congruent quantities, the sums are equal.

Addition of Segments

Addition of Angles

Writing Proofs

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Example 3:

Example 4:

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You Try!Given: KR ≅ SO RM ≅ PS

Prove: KM ≅OP

Example 5:

You Try!

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Summary

Write a valid conclusion for each.

1. Fdfdf

Conclusion: _______________

2. Given: ∡SRT ∡VRY

Conclusion: _______________

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V Y

R

X

TS

Day 4 – Homework

2.

22

3.

23

5. Fddfd

6.

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Day 5 – The Subtraction Postulate

Subtraction of Segments

Example 1:

Example 2:

Subtraction of Angles

Example 3:

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Writing Proofs

Example 4:

Example 5: Given: ∡STE ∡WET ∡STW ∡WES

Prove: ∡WTE ∡SET

26

E

W

T

S

Example 6:

Example 7: Given: ∡TOB ∡WOM

Prove: ∡TOM ∡WOB

27

B W

O

X

MT

SUMMARY

1.

REASON:

2.

REASON:

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Day 5 – HW

1.

29

3.

4.

30

5.

6.

31

Day 6 - Applying the Multiplication/Division Postulate

Writing Proofs

Look for key words like: midpoint, bisect, and trisect The Multiplication property is used when the segments or

angles in the given are less than those in the conclusion! * Small to Big*

32

Ex 1:

33

Writing Proofs Look for key words like: midpoint, bisect, and trisect The Division property is used when the segments or angles in

the given are greater than those in the conclusion!* Big to Small*

Example 2:

34

Summary

Look for key words like: midpoint, bisect, and trisect The Multiplication property is used when the segments or

angles in the given are less than those in the conclusion!* Small to Big*

The Division property is used when the segments or angles in the given are greater than those in the conclusion!

* Big to Small*

1.

REASON:

2. 2.

REASON:

35

Day 6 – HW

1.

2.

36

3.

4.

37

5.

38

6. *Challenge*

39

Day 7 - REVIEW: Proofs with Postulates

1.

2. Given: AC¿ BC BC¿ BA

Prove: AC¿ BA

40

AC

B

3. Given: TS¿ YX , SR¿ XR Prove: TR¿ YR

4. Given: ∡1¿∡2, ∡3¿∡4 Prove: ∡CDA¿∡CBA

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T RS

YX

5. Given: AD≃AE ,DB≃EC Prove: AB≃AC , ∆ABC is isosceles

6. Given: AE≃BD Prove: AD≃BE

42

DX

B

A

C

E

D

X

A

C

BE

7.

43

8.

44

9.

45

Day 8 - REVIEW: Proofs with Postulates

46

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