Unit 5.5

2. GEOMETRY
4-5 Using indirect reasoning
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

3. Writing a Two-Column Proof from a Plan
Use the given plan to write a two-column proof.
Given: Ð1 and Ð2 are supplementary, and
Ð1 @ Ð3
Prove: Ð3 and Ð2 are supplementary.
Plan: Use the definitions of supplementary and congruent
angles and substitution to show that mÐ3 + mÐ2 = 180°.
By the definition of supplementary angles, Ð3 and Ð2 are
supplementary.

4. Writing a Two-Column Proof : Continued
Statements Reasons
Ð1 and Ð2 are supplementary.
Ð1 @ Ð3
1. 1.
2. 2. .
3. . 3.
4. 4.
5. 5.
Given
mÐ1 + mÐ2 =
180°
Def. of supp. Ðs
mÐ1 = mÐ3
mÐ3 + mÐ2 =
180°
Ð3 and Ð2 are supplementary
Def. of @ Ðs
Subst.
Def. of supp. Ðs

5. TEACH! Writing a Two-Column Proof
Use the given plan to write a two-column proof if one
case of Congruent Complements Theorem.
Given: Ð1 and Ð2 are complementary, and
Ð2 and Ð3 are complementary.
Prove: Ð1 @ Ð3
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 Ð1 @ Ð3.

6. TEACH! Continued
Statements Reasons
Ð1 and Ð2 are complementary.
Ð2 and Ð3 are complementary.
1. 1.
2. 2. .
3. . 3.
4. 4.
5. 5.
6. 6.
Given
mÐ1 + mÐ2 = 90°
mÐ2 + mÐ3 = 90°
Def. of comp. Ðs
mÐ1 + mÐ2 = mÐ2 + mÐ3
mÐ2 = mÐ2
mÐ1 = mÐ3
Subst.
Reflex. Prop. of =
Subtr. Prop. of =
Ð1 @ Ð3 Def. of @ Ðs

7. 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.

8. 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
Btheagnin $ b2y5 .assuming that the opposite is true.
That is assume that neither item costs more
tThhains m$2e5a.ns 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.

9. 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.

10. 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

11. Write an indirect proof:
LMN
LMN
D
D
DLMN
Given:
Prove: has at most one right angle.
Indirect proof:
Assume has more than one right angle.
That is assume Ð L a n d Ð M are both right angles.

12. Write an indirect proof:
LMN
LMN
D
D
Given:
Prove: has at most one right angle.
ÐL and ÐM
If are both right angles, then
mÐL=mÐM =90o
According to the Triangle Angle Sum Theorem,.
mÐL+mÐM +mÐN =180o
By substitution: 90o+90o +mÐN =180o
Solving leaves: mÐN =0o

13. Write an indirect proof:
LMN
LMN
D
D
Given:
Prove: has at most one right angle.
mÐN =0o
If: , This means that there is no triangle
LMN. Which contradicts the given statement.
ÐL and ÐM
So the assumption that are both
right angles must be false.
Therefore DLMN has at most one right angle.

14. Lesson Quiz: Part I
Solve each equation. Write a justification for
each step.
1.

15. Lesson Quiz: Part II
Solve each equation. Write a justification for
each step.
2. 6r – 3 = –2(r + 1)

16. Lesson Quiz: Part III
Identify the property that justifies each
statement.
3. x = y and y = z, so x = z.
4. ÐDEF @ ÐDEF
5. AB @ CD, so CD @ AB.

17. Lesson Quiz: Part I
Solve each equation. Write a justification for
each step.
1.
Given
z – 5 = –12 Mult. Prop. of =
z = –7 Add. Prop. of =

18. 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 =

19. Lesson Quiz: Part III
Identify the property that justifies each
statement.
3. x = y and y = z, so x = z.
4. ÐDEF @ ÐDEF
5. AB @ CD, so CD @ AB.
Trans. Prop. of =
Reflex. Prop. of @
Sym. Prop. of @

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