Download Indirect Reasoning

Definition
Consider all the possibilities then prove all but one true.
Writing an indirect prove.
 Find the conclusion and assume that the opposite of what you want to
prove is true.
Use logical reasoning to reach a contradiction of an earlier statement,
such as the given information or a theorem. Then state that the
assumption you made was false.
 State that what you wanted to prove must be true.
Ex. 1
Given: ∆ PQR with obtuse ∠Q.
Prove: m ∠ P 90
Writing an indirect prove.
 Assume (opposite) m ∠ P= 90.
If m ∠ P= 90 then by Tri. Angle Sum Thm m ∠ P + m ∠ Q + m ∠ R =
180, by substitution and given 90 + m ∠ Q + m ∠ R >. This means m ∠ P
90 because it contradicts the tri. angle sum thm.
 ∴An obtuse triangle cannot have a right angle..
Ex. 2
Given: The sum of the measures of the interior ∠’s of a polygon is 900.Q.
Prove: The polygon is not a hexagon.
Writing an indirect prove.
 Assume (opposite) the polygon is a hexagon.
If the polygon is a hexagon then by sum of the angles in a polygon
theorem n-2(180)= 720. This contradicts given statement.
 ∴The polygon is not a hexagon.
Example 1-1 Use Indirect reasoning to prove
statements
( Opp)
Assume that if Andre spends more than $35 to buy
2 items at the sporting good store, then no item
costs more than $17.50.
( Logic)
If neither item costs more than $17.50 then he
spends less than or equal to $35. This contradicts
the given statement.
(Truth)
∴If Andre spends more than $35 to buy 2 items at
the sporting good store, then at least one item
costs more than $17.50.
Example 1-2 Use Indirect reasoning to prove
statements
( Opp)
Assume that if LaShonda spends more than $15 to
buy 3 books at the local book store, then no item
costs more than $5.
( Logic)
If no item costs more than $5 then she spends less
than or equal to $15. This contradicts the given
statement.
(Truth)
∴If LaShonda spends more than $15 to buy 3 books
at the local book store, then at least one item
costs more than $5.
Example 2. 6-11 Write the 1st step in Indirect proof
(opposite)
Assume that ∠𝐴 ≇ ∠𝐵
Assume that △ 𝐷𝐸𝐹 𝒊𝒔 𝒏𝒐𝒕𝒆𝒒𝒖𝒊𝒍𝒂𝒕𝒆𝒓𝒂𝒍
Assume that 𝑊𝑋 𝒊𝒔 𝒏𝒐𝒕 ∥ 𝑌𝑍
Assume that 𝑖𝑠 𝒏𝒐𝒕 𝒔𝒏𝒐𝒘𝒊𝒏𝒈 𝑜𝑢𝑡𝑠𝑖𝑑𝑒
Assume that 𝐴𝐵𝐶𝐷 𝒊𝒔 𝒏𝒐𝒕 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒
Assume that 𝑚 ∠ 𝐴 ≥ 𝟏𝟑𝟓
Example 2. 12-13 Identify Contradiction
I & II
I & III
Example 2. 14-15
Today is not Tuesday.
 Assume that △ ABC is a scalene.
 Definition of scalene is all 3 sides have
different measures, this contradicts the
given diagram..
 ∴ △ ABC is not scalene