Download Chapter 5.5

Chapter 5: Properties and
Attributes of Triangles
5-1 Perpendicular and Angle Bisectors
5-2 Bisectors of Triangles
5-3 Medians and Altitudes of Triangles
5-4 Triangle Midsegment Theorem
5-5 Indirect Proof and Inequalities
5-6 Inequalities in Two triangles
5-7 The Pythagorean Theorem
5-8 Applying Special Right Triangles
5-5 Indirect Proof and
Inequalities in One Triangle
!
So far we have written only DIRECT Proofs. We did these
with logical arguments.
true
hypothesis
“If”
build
logical
argument
show
conclusion
is true
“then”
5-5 Indirect Proof and
Inequalities in One Triangle
!
INDIRECT Proof or “Proof by Contradiction”
Identify
conjecture
Assume
conclusion
“false”
use logic
to show
contradiction
conclude
conjecture
is true
5-5 Indirect Proof and
Inequalities in One Triangle
!
Given: Any triangle
Prove: A triangle cannot have two obtuse angles.
!
1.) Identify the conjecture to be proven:
!
If it is a triangle, then it cannot have two obtuse
angles.
!
5-5 Indirect Proof and
Inequalities in One Triangle
!
Given: Any triangle
Prove: A triangle cannot have two obtuse angles.
!
2.) Assume the opposite (negation) of the conclusion is true.
!
!
Assume temporarily a triangle CAN have two
obtuse angles.
5-5 Indirect Proof and
Inequalities in One Triangle
!
!
!
Given: Any triangle
Prove: A triangle cannot have two obtuse angles.
3.) Use direct reasoning to show that the assumption leads
to a contradiction.
! Assume temporarily that a triangle HAS two obtuse
angles. The measure of any obtuse angle is greater
than 90.
! So the sum of any two obtuse angles is greater than
180; which is a CONTRADICTION to the Triangle Sum
Theorem. So my assumption is false.
5-5 Indirect Proof and
Inequalities in One Triangle
!
!
Given: Any triangle
Prove: A triangle cannot have two obtuse angles.
!
!
4.) Conclude that since the assumption is false, the original
conjecture must be true.
! …So my assumption is false, a triangle cannot have
two obtuse angles.
5-5 Indirect Proof and
Inequalities in One Triangle
!
!
!
!
!
!
!
Given: Any triangle
Prove: A triangle cannot have two obtuse angles.
THE FINAL PROOF
1) Conjecture: A triangle cannot have two obtuse angles.
2) Assume, temporarily, a triangle CAN have two obtuse angles.
3) The measure of any obtuse angle is greater than 90. The sum of
any two obtuse angles is greater than 180. This is a
CONTRADICTION to the theorem that states that the sum of the
angles of any triangle = 180.
4) Therefore, my assumption is false. A triangle cannot have two
obtuse angles.
5-5 Indirect Proof and
Inequalities in One Triangle
!
INDIRECT Proof or “Proof by Contradiction”
Identify
conjecture
Assume
conjecture
“false”
use logic
to show
contradiction
conclude
conjecture
is true
5-5 Indirect Proof and
Inequalities in One Triangle
!
Which angle is largest?
5-5 Indirect Proof and
Inequalities in One Triangle
!
A triangle can be formed by 3
segments, but not every set of three
segments.
!
!
!
!
So how are you supposed to know?
5-5 Indirect Proof and
Inequalities in One Triangle
!
Theorem 5-5-3: Triangle Inequality
Theorem
!
The sum of any two side lengths of a triangle
is greater than the third side length.
A
!
AB + BC > AC
! BC + AC > AB
! AC + AB > BC
!
C
B