GEOMETRY

4-5 Using indirect reasoning

Warm UpComplete 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

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

Writing a Two-Column Proof from a Plan

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 m3 + m2 = 180°. By the definition of supplementary angles, 3 and 2 are supplementary.

Writing a Two-Column Proof : Continued

Statements Reasons

1. 1.

2. 2. .

3. . 3.

4. 4.

5. 5.

1 and 2 are supplementary.

1 3

Given

m1 + m2 = 180° Def. of supp. s

m1 = m3

m3 + m2 = 180°

3 and 2 are supplementary

Def. of s

Subst.

Def. of supp. s

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

TEACH! Writing a Two-Column Proof

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

Statements Reasons

1. 1.

2. 2. .

3. . 3.

4. 4.

5. 5.

6. 6.

1 and 2 are complementary.

2 and 3 are complementary.

Given

m1 + m2 = 90° m2 + m3 = 90°

Def. of comp. s

m1 + m2 = m2 + m3

m2 = m2

m1 = m3

Subst.

Reflex. Prop. of =

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.

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.

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

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.

Given:

Prove: has at most one right angle.

LMN

LMN

LMN

That is assume are both right angles. and L M

Write an indirect proof:

If are both right angles, then

Given:

Prove: has at most one right angle.

LMN

LMN

According to the Triangle Angle Sum Theorem,.

and L M =m 90om L M

+m 180om L M m N By substitution: 90 +90 180o o om N Solving leaves: 0om N

Write an indirect proof:Given:

Prove: has at most one right angle.

LMN

LMN

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

0om N

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

and L M

Therefore has at most one right angle.LMN

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.

Lesson Quiz: Part I

Solve each equation. Write a justification for each step.

1.

z – 5 = –12 Mult. Prop. of =

z = –7 Add. Prop. of =

Given

Lesson Quiz: Part II

Solve each equation. Write a justification for each step.

2. 6r – 3 = –2(r + 1)

Given

6r – 3 = –2r – 2

8r – 3 = –2

Distrib. Prop.

Add. Prop. of =

6r – 3 = –2(r + 1)

8r = 1 Add. Prop. of =

Div. Prop. of =

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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