· Web viewExample 2: Find an equation of a line in slope intercept form given the following, then write in standard form. Passes through the points (6, 2) and (7, 0) Passes through

Secondary 2 Chapter 2

Secondary II Chapter 2 – Introduction to Proof

2014/2015

Date Section Assignment Concept

A: 8/25B: 8/26

School Starts (Advisory/Assemblies, etc)Disclosures

A: 8/27B: 8/28

Pre-TestReview Assignment #1

A: 8/29 Review Assignment #2

9/1/14 Labor Day

B: 9/2 Review Assignment #2

A: 9/3B: 9/4 Review Assignment #3

A: 9/5B: 9/8 2.1 - Worksheet 2.1 Foundations for Proof

A: 9/9B: 9/10 2.2 - Worksheet 2.2 Special Angles and Postulates

A: 9/11B: 9/12 2.3 - Worksheet 2.3

Paragraph Proof, Two-Column Proof, Construction Proof, and Flow Chart Proof

A: 9/15B: 9/16

2.4 & 2.5 - Worksheet 2.4 & 2.5

Angle Postulates and TheoremsParallel Line Converse Theorems

A: 9/17B: 9/18 Review

A: 9/19B: 9/22 TEST

Late and absent work will be due on the day of the review (absences must be excused). The review assignment must be turned in on test day. All required work must be complete to get the curve on the test.

Remember, you are still required to take the test on the scheduled day even if you miss the review, so come prepared. If you are absent on test day, you will be required to take the test in class the day you return. You will not receive the curve on the test if you are absent on test day unless you take the test prior to your absence.

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Review Notes

Day1 Review Notes: GFC, LCM, and Prime Factorization

Example 1: Find the prime factorization for the following numbers.

a) 55 b) 13 c) 162

Example 2: Find the LCM for each set of numbers.

a) 9 & 18 b) 3 & 7 c) 6 & 12 & 36

Example 3: Find the GCF for each set of numbers.

a) 9 & 18 b) 3 & 7 c) 6 & 12 & 36

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Additional Notes

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Day 2 Review Notes: Lines and System of Equations

Define the following terms:

1. Slope-

2. Standard form of a line-

3. Slope-intercept form of a line-

4. Point-slope form of a line-

5. Parallel lines-

6. Perpendicular lines-

7. x- and y- intercepts-

Example 1: Find the slope between each pair of points.

a) (-2, 5) and (3, -1) b) (9, 7) and (6, 7) c) (3, 4) and (3, -8)

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Example 2: Find an equation of a line in slope intercept form given the following, then write in standard form.

a) Passes through the points (6, 2) and (7, 0)

b) Passes through the point (2, 1) and is parallel to 4x-2y = 3

c) Passes through the point (2, 1) and is perpendicular to 4x-2y = 3

Example 3: Cam prints and sells t-shirts for $14.99 each. Each month 5 t-shirts are misprinted and cannot be sold. Write a linear equation that represents the total amount Cam earns each month selling t-shirts taking into account the value of the t-shirts that cannot be sold.

Example 4: A florist sells carnations for $10.99 a dozen and lilies for $12.99 a dozen. During a weekend sale, the florist’s goal is to earn $650. Write an equation that represents the total amount the florist would like to earn selling carnations and lilies during the sale.

5

x

y


Example 5: Find the x- and y- intercepts of the equation, and graph.

a) 2x + 3y = 10 b) y = 2

Define the following:

1. System of linear equations-

2. Consistent systems-

3. Inconsistent systems-

Example 6: Chen starts his own lawn mowing business. He initially spends $180 on a new mower. For each yard he mows, he receives $20 and spends $4 on gas. Write a system of linear equations to represent each problem. Define each variable. Graph the system of equations, label the axes.

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x

y


Example 7: Solve by substitution. Determine if the system is consistent or inconsistent.

a) b)

Example 8: Solve by elimination. Determine if the system is consistent or inconsistent.

a) b)

Example 9: Write a system of equations to represent each problem situation. Solve the system and explain what your solution means in context of the problem.

The Pizza Barn sells one customer 3 large pepperoni pizzas and 2 orders of breadsticks for $30. They sell another customer 4 large pepperoni pizzas and 3 orders of breadsticks for $41. How much does the Pizza Barn charge for each pizza and each order of breadsticks?

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Additional Notes

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Day 3 Review Notes: Radicals and Exponents

Example 1: Simplify each expression.

a) c)

b) √27 d)

Example 2: Write each radical as a power or each power as a radical.

a) √3 b) c) d) 3√ae)

Example 3: Solve the exponential equation.

a)3x=243

b) c)

9

x

y

x

y


Remember the parent graph y=bx and its properties

Example 4: Describe the transformation(s) from f(x) to g(x).

a) f ( x )=2x b) f ( x )=2x c) f ( x )=2x

g ( x )=2x+3 g ( x )=2x+3 g ( x )=−2x

Example 5: Complete each table and graph the function. Identify the x-intercept, y-intercepts, asymptote, domain, range, and interval(s) of increase or decrease for the function.

a)f ( x )=3x

x-intercept:

y-intercept:

asymptote:

domain:

range:

intervals:

10

x f(x)

-2

-1

0

1

2

x

y

x

y


b)f ( x )=−2x

x-intercept:

y-intercept:

asymptote:

domain:

range:

intervals:

c) f ( x )=4x−2

x-intercept:

y-intercept:

asymptote:

domain:

range:

intervals:

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x f(x)

-2

-1

0

1

2

x f(x)

-2

-1

0

1

2


Additional Notes

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Chapter 2: Introduction to Proof

2.1 – Foundations of Proof

The ability to use information to reason and make conclusions is very important in life an in mathematics. There are two common methods of reasoning. You can construct the name for each method of reasoning using you knowledge of prefixes, root words, and suffixes.

1. Form a word that means “the act of thinking down from.”

2. Form a word that means “the act of thinking toward or up to.”

Define:

Induction/ Inductive reasoning:

Deduction/Deductive reasoning:

Example 1: Determine whether inductive reasoning or deductive reasoning is used in each situation. Then determine whether the conclusion is correct and explain your reasoning.

1. Jose is shown the first six numbers of a series of numbers 7,11,15,19,23,27. He concludes that the general rule for the serious of numbers is an=4 n+3.

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2. Sarah is told all berries from plants are poisonous. She sees a plant with berries on it and concludes they are poisonous.

3. Mark has been told that lightning never strikes twice in the same place. During a lightning storm, he sees a tree struck by lightning and goes to stand next to it, convinced that it is the safest place to be.

~ A problem situation can provide you with a great deal of information that you can use to make conclusions. It is important to identify specific and general information in a problem situation to reach appropriate conclusions. Some information may be irrelevant to reach the appropriate conclusion~

Example 2: Identify the specific information, the general information, and the conclusion for each problem.

1. Connor read an article that claimed that tobacco use greatly increases the risk of getting cancer. He then noticed that his neighbor Matilda smokes. Connor is concerned that Matilda has a high risk of getting cancer.

Specific information: __________________________________________________________

General Information: __________________________________________________________

Conclusion: __________________________________________________________________

2. Molly returned from a trip to England and tells you, “It rains every day in England!” She explains that it rained each of the five days she was there.

Specific information: __________________________________________________________

General Information: __________________________________________________________

Conclusion: __________________________________________________________________

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A conditional statement is a statement that can be written in the form “If p, then q.” This form is the propositional form of a conditional statement. It can also be written using symbols as p → q, which is read as “p implies q.” The variables p and q are propositional variables. The hypothesis of a conditional statement is the variable p. The conclusion of a conditional statement is the variable q.

Example 3: Write each statement in propositional form. Then identify the hypotenuse and the conclusion.

1. The measure of an angle is 90ᵒ. So, the angle is a right angle.

Propositional Form:__________________________________________

Hypothesis: ________________________________________________

Conclusion:________________________________________________

2. Two lines are not on the same plane. So, the points are collinear.

Propositional Form:__________________________________________

Hypothesis: ________________________________________________

Conclusion:________________________________________________

A Hypothesis can also be considered as the Given to a proof, and the Conclusion can be considered as the Prove of a proof.

Example 4: For each conditional statement write the hypothesis as the “Given” and the conclusion as the “Prove.”

Given:________________________________

Prove:_________________________________

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Additional Notes

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2.2 – Special Angles and Postulates

Two angles are supplementary angles if the sum of their angle measures is equal to 180ᵒ

Two angles are complementary angles if the sum of their angle measures is equal to 90ᵒ

Adjacent Angles are angles that share a side

A Linear Pair of angles are two adjacent angles that have noncommon sides that form a line. Angles that form a linear pair add up to 180ᵒ

Vertical Angles are two nonadjacent angles that are formed by two intersecting lines. Vertical angles are congruent.

Example 1: Draw a few examples of each type of angles and nonexamples:

Supplementary Angles Complementary Angles

Adjacent Angles Linear Pair

Vertical Angles

Example 2: Decide if each set of angles are complementary, supplementary or neither. Justify your answer with work.

a) b) c)

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Example 3: Name the angle(s) that are supplementary

to in the picture at right.

Example 4: Find x

a) b)

c) d)

Example 5: Given: and find:

a)

b)

c)

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Example 6:

a) The measure of the supplement of an angle is one fourth the measure of the angle. What is the angle?

b) The measure of the complement of an angle is one fifth the measure of the angle. What is the measure of each angle?

Example 7: List all the pairs of vertical angles, then find the value of all the measures given m<1=60ᵒ

Example 8: Solve for x

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A postulate is a statement that is accepted without proof.

A theorem is a statement that can be proven.

The Linear Pair Postulate states: “If two angles form a linear pair, then the angles are supplementary .”

The Segment Addition Postulate states: “If point B is on AC and between points A and C, then AB + BC = AC .”

The Angle Addition Postulate states: “If point D lies in the interior of ∠ ABC, then m∠ ABD+m∠DBC=m∠ABC

Example 9: Write the postulate that confirms each statement.

a) Angles GFH and KFH are b) mRS+mST=mRT

c) m∠WXZ+m∠ZXY=mWXY d) m∠1+m∠2=180 °

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Additional Notes

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2.3 – Special Angles and Postulates

Many properties of real numbers can be applied in geometry. These properties are important when making conjectures and proving new theorems.

Here is a list of properties that will be helpful to know when proving.

Addition Property of Equality: If a, b, and c are real numbers and a = b, then a + c = b + c.

Examples:

Subtraction Property of Equality: If a, b, and c are real numbers and a = b, then a – c = b – c.

Examples:

Reflexive Property: If a is a real number, then a = a

Examples:

Substitution Property: If a and b are real numbers and a = b, then a can be substituted for b.

Examples:

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Transitive Property: If a, b, and c are real numbers and a = b, and b = c, then a = c.

Examples:

Additional Notes

Example 1: Identify the property demonstrated in each example.

a)

b)

c)

Example 2: Rewrite each conditional statement by separating the hypothesis and conclusion. The

hypothesis becomes the “Given” and the conclusion becomes the “Prove”

Conditional statement: If <2≅<1, then <2≅<3.

Given:

Prove:

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A Proof is a logical serious of statements and corresponding reasons that starts with a hypotenuse and arrives at a conclusion. In this course you will use three different kinds of proofs.

Flow Chart Proof:

Two Column Proof:

Paragraph Proof:

Example 3: Given the flow chart write as a two-column proof, prove the Vertical Angle Theorem.

Statement Reason

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Example 4: Given the two-column proof write as a paragraph proof

Prove: Congruent Supplement Theorem

Example 5: Prove the statement using a flow chart

Given: <ABC and <XYZ are straight angles

Prove: <ABC≅<XYZ

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Example 6: Prove the statement using a two column proof

Given: ¿DEG≅<HEF

Prove: ¿DEH ≅<GEF

Statement Reason

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Additional Notes

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2.4/2.5 – Angle postulates and Theorems, Parallel Line Converse Theorems

In this section we will learn more postulates and theorems to help use prove and understand angle measures better.

Use the following diagram to help understand the postulates and theorems

If two parallel lines are intersected by a transversal, then:

• corresponding angles are congruent .

• alternate interior angles are congruent .

• alternate exterior angles are congruent .

• same-side interior angles are supplementary .

• same-side exterior angles are supplementary .

Each of these relationships is represented by a postulate or theorem.

• Corresponding Angle Postulate: If two parallel lines are intersected by a transversal, then corresponding angles are congruent .

• Alternate Interior Angle Theorem: If two parallel lines are intersected by a transversal, then alternate interior angles are congruent .

• Alternate Exterior Angle Theorem: If two parallel lines are intersected by a transversal, then alternate exterior angles are congruent .

• Same-Side Interior Angle Theorem: If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are supplementary.

• Same-Side Exterior Angle Theorem: If two parallel lines are intersected by a transversal, then exterior angles on the same side of the transversal are supplementary

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Example 1: Given the following figure, list all examples of each relationship.

a) Vertical angles:_________________________________________________________________

b) Alternate Interior angles:_________________________________________________________

c) Alternate Exterior angles:_________________________________________________________

d) Same-side exterior angles:________________________________________________________

e) Same-side interior angles:________________________________________________________

f) Corresponding Angles:___________________________________________________________

g) Linear Pairs:___________________________________________________________________

Example 2: Given m<4= 37ᵒ find all the other angle measures.

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Example 3: Use the following picture to solve for the given values.

a) Given m<1 = x+70 and m<2 = x find x and the rest of the angle measures.

b) Given m<5 = 2x and the m<2 = x+50 find x and the rest of the measures

Define Converse:

Example 4: For each theorem, identify the hypothesis, p and conclusion, q. Then write the converse.

a)

b)

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c)

Each of the theorems and postulates introduced today also have converses. It is important to understand the theorem and its converse.

Corresponding Angle Converse Postulate: If two lines intersected by a transversal form congruent corresponding angles, then the lines are parallel.

Alternate Interior Angle Converse Theorem: If two lines intersected by a transversal form congruent alternate interior angles, then the lines are parallel.

Alternate Exterior Angle Converse Theorem: If two lines intersected by a transversal form congruent alternate exterior angles, then the lines are parallel.

Same-Side Interior Angle Converse Theorem: If two lines intersected by a transversal form supplementary same-side interior angles, then the lines are parallel.

Same-Side Exterior Angle Converse Theorem: If two lines intersected by a transversal form supplementary same-side exterior angles, then the lines are parallel.

Example 5: Use the following diagram to answer the questions

a) Which theorem or postulate would use ∠2≅∠7 to justify line p is parallel to line r?

b) Which theorem or postulate would use ∠4 ≅∠5 to justify line p is parallel to line r?

c) Which theorem or postulate would use ∠1≅∠5 to justify line p is parallel to line r?

d) Which theorem or postulate would use m∠1+m∠7=180° to justify line p is parallel to line r?

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Example 6: Prove the Alternate Interior Angle: If two parallel lines are intersected by a transversal, then alternate interior angles are congruent.”

Example 7: Prove the Alternate Exterior Angle Converse: “If two lines intersected by a transversal form congruent alternate exterior angles, then the lines are parallel.”

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Additional Notes

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Documents

· Web viewExample 2: Find an equation of a line in slope intercept form given the following, then write in standard form. Passes through the points (6, 2) and (7, 0) Passes through