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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 m3 + m2 = 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

m1+ m2 = 180°

Def. of supp. s

m1= m3

Def. of s

Subst.

m3+ m2 = 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

m1+ m2 = 90° m2+ m3 = 90°

Def. of comp. s

m1+ m2 = m2+ m3

Subst.

Reflex. Prop. of =

m2= m2

m1 = m3

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.

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

Indirect proof:

Assume has more than one right angle.

That is assume are both right angles.

If are both right angles, then

According to the Triangle Angle Sum Theorem,.

By substitution:

Solving leaves:

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.

Identify the property that justifies each statement.

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

4. DEF DEF

5. ABCD, so CDAB.

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.

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)

Identify the property that justifies each statement.

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

4. DEF DEF

5. ABCD, so CDAB.

Trans. Prop. of =

Reflex. Prop. of

Sym. Prop. of

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