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GEOMETRY UNIT 1

WORKBOOK

CHAPTER 2 Reasoning and Proof

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: Using postulates and diagrams, make valid conclusions about points, lines, and planes.

I) Reminder:

• Rules that are accepted without proof are called _________________ or _________________.

• Rules that are proved are called _________________.

II) Point, Line, Plane Postulates:

a) Write the answers to the questions on the Partner Investigation below. 1)

2)

3)

4)

5)

6)

7)

b) Use your observations from the Partner Investigation to complete the following.

• Through any two points you can only draw ______ line(s). (#1)

• To draw a line, you must have at least ______ points. (#2)

• If two lines intersect, then their intersection is exactly ______ point(s). (#4)

• You can draw exactly ______ plane(s) through any three noncollinear points. (#5)

• To make a plane, you need at least ______ noncollinear point(s). (#3)

• If two points lie in a plane, then the line going through them also lies in the ___________.

• If two planes intersect, then their intersection is a ______________. (#7)

Notes 2.5

Hint: Look at your answers from these problems above.

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Write a proof in the following example.

1) GIVEN: ,A B B C≅ ≅ PROVE: A C≅

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Take notes on pg. 136 in textbooks (key concept) & fill in the blanks below

Example 1 provides an example of an algebraic proof. Write a proof for Example 2. Show each step on a different line.

1) 2(5 – 3a) – 4(a + 7) = 92 2) -4(11x + 2) = 80

In your own words, what does “simplify/combine like terms” mean?

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Take notes on pg. 144-145 in textbook on Segment Addition Postulate & Example 1

When do you think “definition of congruent segments” is used? What is always the first reason? What is always the last statement?

Draw a picture

Fill in the blanks

Definition of congruent segments

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Warm-Up Given: WY = YZ YZ XZ≅ XZ WX≅ Prove: WXWY ≅

Quick Notes Definition of Midpoint: . Given: AB DE≅ B is the midpoint of .AC E is the midpoint of .DF Prove: BC EF≅

Statements Reasons 1. 2. AB = DE 3.

1. Given 2. 3. Given

4.

4. Definition of Midpoint

5.

5. Given

6.

6. Definition of Midpoint

7. DE = BC

7.

8. BC = EF 9.

8. 9.

Statements Reasons 1. WY = YZ

2. 𝑊𝑊����� ≅ 𝑊𝑌����

3. 𝑊𝑌���� ≅ 𝑋𝑌����

4. 𝑊𝑊����� ≅ 𝑋𝑌����

5. 𝑋𝑌���� ≅ 𝑊𝑋����� 6.

1.

2.

3.

4.

5.

6.

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In-Class Practice Complete each proof. 1. Given: BC = DE Prove: AB + DE = AC Proof:

Statements Reasons 1. BC = DE 1. _____________ 2. _____________ 2. Segment Addition Postulate

3. AB + DE = AC 3. _____________

2. Given: X is the midpoint of =MN and MX RX Prove: XN ≅ RX

3. Given: GD BC≅ BC FH≅ FH AE≅ Prove: AE GD≅

You can now fill in #’s 1 – 7 on proof reasons sheet!!

Statements Reasons

1. X is the midpoint of MN 1.

2. XN = MX 2.

3. MX = RX 3.

4. XN = RX 5. XN ≅ RX

4. 5.

Statements Reasons 1.

2.

3.

4.

5.

6.

1. Given

2. Given

3. Transitive Property

4. Given

5. Transitive

6.

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Take notes on the following topics beginning on page 151 of your textbook. Draw pictures to help. *Angle Addition Postulate: Supplement Theorem: *What does supplementary mean? Complement Theorem: *What does complementary mean? Congruent Supplements Theorem: Congruent Complements Theorem: *Vertical Angles Theorem: *Indicates reasons that are often used in proofs

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Read through this section, highlight/underline important pieces, and write questions you have.

When thinking about “definition of congruent segments,” when do you think “definition of congruent angles” is used? This is not explicitly in this section. Just think about it using your reasoning skills.

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Warm-Up Write a proof for the following: Given: ∡3 ≅ ∡2 Prove: ∡3 ≅ ∡6

Statements Reasons

1. ∡3 ≅ ∡2 2. ∡2 ≅ ∡6 3. ∡3 ≅ ∡6

Quick Notes Given: ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove: ∠3 and ∠4 are congruent.

In-Class Practice For numbers 1 & 2, find the value of x and name the theorems that justify your work. 1. m∠1 = (x + 10)° 2. m∠6 = (7x – 24)° m∠2 = (3x + 18)° m∠7 = (5x + 14)° 1. x = __________________ 2. x = _________________ Theorem: __________________ Theorem: _________________

Statements Reasons 1.

2. ∠1 and ∠4 are supplementary

3. m∠3 + m∠1 = 180°

4. m∠3 and m∠1 are supplementary 5.

1.

2. 3.

4.

5.

2 3

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3. Write a proof for the following: Given: m∡4=120o, ∡2 ≅ ∡5, ∡5 ≅ ∡4 Prove: m∡2=120o

Statements Reason 1) m∡4=120o 2) ∡2 ≅ ∡5; ∡5 ≅ ∡4 3) ∡2 ≅ ∡4 4) m∡2 = m∡4 4. Given: ∠6 ≅ ∠5 Prove: ∠4 ≅ ∠7 5. Write a proof for the following: Given: ∡1 ≅ ∡5; ∡4 and ∡5 are a linear pair Prove: ∡1 is supplementary to ∡4 Statements Reasons 1. ∡1 ≅ ∡5 2. 𝑚∡1 = 𝑚∡5 3. ∡4 and ∡5 are a linear pair 4. ∡4 and ∡5 are supplementary 5. m∡4 + m∡5 = 180o 6. m∡4 + m∡1 = 180o

7. ∡1 is supplementary to ∡4

You can now fill in #’s 11 – 24 on proof reasons sheet!!

Statements Reasons 1. ∠6 ≅ ∠5 2. ∠5 ≅ ∠4 3. ∠6 ≅ ∠4 4. ∠4 ≅ ∠6 5. ∠6 ≅ ∠7 6.

1. 2. 3. 4. 5. 6.

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4 5 6 7

1 2 3

4 5

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Geometry Name _____________________________________ REVIEW 2.5 – 2.8 For numbers 1 – 3, determine if the statement is always (A), sometimes (S), or never (N) true. 1. If two points lie in a plane, then the entire line containing those points lies in that plane. 2. Two lines intersect to form right angles. 3. Three noncollinear points are contained in Plane Y. For numbers 4 – 6, find the measure of the indicated angle and name the theorems that justify your work. 4. If m∠1 = (x + 50)° and m∠2 = (3x – 20)°, find m∠1. m∠1 = _______________ Theorem: ____________________ 5. If m∠1 is twice m∠2, find m∠1. m∠1 = _______________ Theorem: ____________________ 6. If ∠ABC ≅ ∠EFG and m∠ABC = 41, find m∠GFH. m∠GFH = ____________ Theorem: ____________________ For numbers 7 – 12, state the definition, property, postulate, or theorem that justifies each statement. 7. If X is the midpoint of 𝐶𝐷���� , then CX = XD . 8. If ∠K ≅ ∠P and ∠P ≅ ∠T, then ∠K ≅ ∠T. 9. If m∠W + m∠H = 90° and m∠H = 20°, then m∠W + 20 = 90. 10. If ∠A and ∠B are complementary and ∠A and ∠D are complementary, then ∠B ≅ ∠D. 11. If 𝐴𝑇���� ≅ 𝐷𝑅���� then AT = DR. 12. AB + BC = AC

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13. Complete the proof by supplying the missing information

If 2x – 7 = 4, then x = 112

Proof

Statements Reasons

1.

2.

3.

1.

2.

3.

14. Given: M is the midpoint of AB MB BX≅ Prove: AM BX≅

15. Given: Ð1 ≅ Ð2 Ð2 ≅ Ð3 Ð3 ≅ Ð4 Prove: Ð1 ≅ Ð4 Statements Reasons 1. Ð1 ≅ Ð2 2. Ð2 ≅Ð3 3. Ð1≅ Ð3 4. Ð3 ≅ Ð4 5. Ð1 ≅ Ð4

1. 2. 3. 4. 5.

Statements Reasons 1. M is the midpoint of AB 2. AM = MB 3._________________________ 4. .________________________ 5. .________________________

1. _________________________ 2. _________________________ 3. _________________________ 4. Given 5. _________________________

Reason Bank: Addition Property Congruent complements theorem Congruent supplements theorem Definition of complementary angles Definition of congruent angles Definition of congruent segments Definition of midpoint Definition of supplementary angles Distributive Property Division Property Reflexive Property Midpoint Theorem Multiplication Property Segment Addition Postulate Substitution Property Subtraction Property Supplement Theorem Symmetric Property Transitive Property Vertical angles are Congruent

A B M ● ● ●

1 2

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