Transcript 5-3: Indirect Proof and Inequalities

5-3: Indirect Proof and Inequalities
Expectations:
G1.2.2: Construct and justify arguments
and solve multi-step problems involving
side length, angle measure, perimeter and
areas of all types of triangles.
L3.3.2: Construct proofs by contradiction.
Use counterexamples to disprove a
statement.
Indirect Reasoning
Making a conclusion by ruling out all other
possibilities.
Steps in indirect reasoning:
a. Determine all possibilities for an event.
b. Assume a possibility you think is false to be true.
c. Show that this possibility contradicts either the given, or
a postulate, definition or theorem. When this happens it
shows that the possibility cannot be true and can be
eliminated.
d. When all possibilities except 1 have been ruled out, the
only 1 left must be the truth.
When we use indirect reasoning in a proof, we call
it indirect proof or proof by contradiction.
Prove a triangle cannot have 2
right angles.
a. What are all of the possibilities for this
situation?
b. Which possibility should we assume?
Prove a triangle cannot have 2
right angles.
Either a triangle can have 2 right angles or it
cannot. Let’s assume a triangle can have 2 right
angles. Since the third angle of a triangle must
have a positive measure, the sum of the three
angle measures will be greater than 180. This
contradicts the triangle sum theorem. This
means our assumption must be incorrect. This
leaves us with only 1 possibility left - a triangle
cannot have 2 right angles.
State the possibilities and assumption you should
make to start an indirect proof of the statement
below.
Lines m and n intersect at P
State the possibilities and assumption you should
make to start an indirect proof of the statement
below.
Triangle ABC is isosceles
Given: XZ > WX
Prove: ∠W is not ≅ ∠Z
Remember the given is
always true so do not
change the given in your
assumption.
X
W
Y
Z
Given: m is not parallel to n
Prove: ∠1 is not ≅ ∠2
t
m
n
1
2
Inequalities
Defn: Inequality: For any real numbers a and
b, a < b iff there is a positive real number
c such that a + c = b.
Exterior Angle Inequality Theorem
If an angle is an exterior angle of a triangle, then its
measure is greater than either of its remote
interior angles.
Exterior Angle Inequality Theorem
B
m∠BCD > m∠A
m∠BCD > m∠B
A
C
D
Properties of Inequality
For all real numbers a, b, and c:
A) a < b, a = b or a > b. (Comparison property)
B) If a < b and b < c, then a < c.
(Transitive property)
C) If a < b, then a + c < b + c and if a < b,
then a – c < b – c.
(Addition and Subtraction Properties)
Properties of Inequality
(Multiplication and Division properties)
D. if a < b and c > 0, then ac < bc and a < b.
c
c
and if a < b and c < 0, then ac > bc and a > b.
c
c
ACT/MME
Milo plans to prove that no isosceles right triangle has
whole-unit measurements for all three sides. To
construct a proof by contradiction, which should be
Milo's first step?
A. Show that a2 + b2 = c2.
B. Assume that such a triangle exists.
C. Pick three whole numbers that form a Pythagorean
Triple.
D. Assume no triangle has whole-unit measurements for all
three sides.
Assignment
Page 256,
# 17 – 27(odds) and 39