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Warm Up. Lesson Presentation. Lesson Quiz. Warm Up Complete each sentence. 1. If the measures of two angles are _____, then the angles are congruent. 2. If two angles form a ________ , then they are supplementary.

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Warm Up

Lesson Presentation

Lesson Quiz


Warm Up

Complete each sentence.

1.If the measures of two angles are _____, then the angles are congruent.

2. If two angles form a ________ , then they are supplementary.

3. If two angles are complementary to the same angle, then the two angles are ________ .

equal

linear pair

congruent


Writing a Two-Column Proof from a Plan

Use the given plan to write a two-column proof.

Given: 1 and 2 are supplementary, and

1  3

Prove: 3 and 2 are supplementary.

Plan: Use the definitions of supplementary and congruent angles and substitution to show that m3 + m2 = 180°.By the definition of supplementary angles, 3 and 2are supplementary.


Writing a Two-Column Proof : Continued

Given

1 and 2 are supplementary.

1  3

m1+ m2 = 180°

Def. of supp. s

m1= m3

Def. of s

Subst.

m3+ m2 = 180°

Def. of supp. s

3 and 2 are supplementary


TEACH! Writing a Two-Column Proof

Use the given plan to write a two-column proof if one case of Congruent Complements Theorem.

Given: 1 and 2 are complementary, and

2 and 3 are complementary.

Prove: 1  3

Plan: The measures of complementary angles add to 90° by definition. Use substitution to show that the sums of both pairs are equal. Use the Subtraction Property and the definition of congruent angles to conclude that 1 3.


TEACH! Continued

1 and 2 are complementary.

2 and 3 are complementary.

Given

m1+ m2 = 90° m2+ m3 = 90°

Def. of comp. s

m1+ m2 = m2+ m3

Subst.

Reflex. Prop. of =

m2= m2

m1 = m3

Subtr. Prop. of =

1  3

Def. of  s


Use indirect reasoning to prove:

If Jacky spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25.

Given: the cost of two items is more than $50.

Prove: At least one of the items costs more than $25.

Begin by assuming that the opposite is true. That is assume that neither item costs more than $25.


Use indirect reasoning to prove:

If Jacky spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25.

Given: the cost of two items is more than $50.

Prove: At least one of the items costs more than $25.

Begin by assuming that the opposite is true. That is assume that neither item costs more than $25.

This means that both items cost $25 or less. This means that the two items together cost $50 or less. This contradicts the given information that the amount spent is more than $50. So the assumption that neither items cost more than $25 must be incorrect.


Use indirect reasoning to prove:

If Jacky spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25.

Therefore, at least one of the items costs more than $25.

This means that both items cost $25 or less. This means that the two items together cost $50 or less. This contradicts the given information that the amount spent is more than $50. So the assumption that neither items cost more than $25 must be incorrect.


Writing an indirect proof

Step-1: Assume that the opposite of what you want to prove is true.

Step-2: Use logical reasoning to reach a contradiction to the earlier statement, such as the given information or a theorem. Then state that the assumption you made was false.

Step-3: State that what you wanted to prove must be true


Write an indirect proof:

Indirect proof:

Assume has more than one right angle.

That is assume are both right angles.


Write an indirect proof:

If are both right angles, then

According to the Triangle Angle Sum Theorem,.

By substitution:

Solving leaves:


Write an indirect proof:

If: , This means that there is no triangle LMN. Which contradicts the given statement.

So the assumption that are both right angles must be false.


Lesson Quiz: Part I

Solve each equation. Write a justification for each step.

1.


Lesson Quiz: Part II

Solve each equation. Write a justification for each step.

2.6r – 3 = –2(r + 1)


Lesson Quiz: Part III

Identify the property that justifies each statement.

3. x = y and y = z, so x = z.

4. DEF  DEF

5. ABCD, so CDAB.


Given

z – 5 = –12

Mult. Prop. of =

z = –7

Add. Prop. of =

Lesson Quiz: Part I

Solve each equation. Write a justification for each step.

1.


Given

6r – 3 = –2(r + 1)

6r – 3 = –2r – 2

Distrib. Prop.

Add. Prop. of =

8r – 3 = –2

8r = 1

Add. Prop. of =

Div. Prop. of =

Lesson Quiz: Part II

Solve each equation. Write a justification for each step.

2.6r – 3 = –2(r + 1)


Lesson Quiz: Part III

Identify the property that justifies each statement.

3. x = y and y = z, so x = z.

4. DEF  DEF

5. ABCD, so CDAB.

Trans. Prop. of =

Reflex. Prop. of 

Sym. Prop. of 


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