LESSON PLAN 6
Geometry – Logic and Intro to Proof (Block 4-6), Grades 9-12
Conjectures and Theorems (Conjecture, Proof, Truth Table)
1. Proofs
A proof is a rigorous demonstration that a proposition is true. A direct proof, is a
proof whose conclusion follows directly (after usually some steps) from the given
premises. An indirect proof (or proof by contradiction), is a proof in which we
assume that the conclusion is false, and then we arrive at a contradiction (which
means that the conclusion is thereby true).
2. We assume here that the basic notions of axioms, theorems, etc,
are known (or have been introduced) to the student.
[An excellent reference book on the subject is: Mathematical Ideas, C. Miller, V. Heeren, J.
Hornsby, M. Morrow, and J. Newenhizen, 10th ed, Pearson, 2004.]
3. First we discuss direct proofs:
The following were taken from:
http://www.sparknotes.com/math/geometry3/geometricproofs/problems.html#ex
planation1
Problem :
Given:
Circle C with triangles ABC and DEC.
Chord AB is congruent to chord DE.
Prove:
Triangles ABC and DEC are congruent.
Solution
4. Now we discuss indirect proofs:
The following were taken from:
http://regentsprep.org/Regents/mathb/1e/indirectlesson.htm
When trying to prove a statement is true, it may be
beneficial to ask yourself, "What if this statement was
not true?" and examine what happens. This is the
premise of the Indirect Proof or Proof by
Contradiction.
Indirect Proof:
Assume what you need to prove is false, and
then show that something contradictory
(absurd) happens.
Steps in an Indirect Proof:
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


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Assume that the opposite of what you are trying to prove is true.
From this assumption, see what conclusions can be drawn. These conclusions
must be based upon the assumption and the use of valid statements.
Search for a conclusion that you know is false because it contradicts given or
known information. Oftentimes you will be contradicting a piece of GIVEN
information.
Since your assumption leads to a false conclusion, the assumption must be
false.
If the assumption (which is the opposite of what you are trying to prove) is
false, then you will know that what you are trying to prove must be true.
How to Recognize When
an Indirect Proof is Needed:
Proof by Contradiction
is also known as
reductio ad absurdum
(which from Latin means
reduced to an absurdity).
Generally, the word "not" or the presence of a "not
symbol" (such as the not equal sign
) in a
problem indicates a need for an Indirect Proof.
Example:
(done in a two-column format)
In the accompanying diagram,
is not isosceles.
Prove that if altitude
is drawn, it will not bisect
.
In this example, we must first
clearly indicate the GIVEN
and the PROVE.
Given:
Prove:
STATEMENTS
1.
REASONS
1. Given
2. Assume
(Remember to assume the opposite of the
PROVE.)
2. Assumption leading to a
contradiction.
3.
3. Bisector of a segment divides the
segment at its midpoint.
4.
4. Midpoint divides a segment into two
congruent segments.
5.
5. The altitude of a triangle is a line
segment extending from any vertex
of a triangle perpendicular to the
line containing the opposite side.
6.
6. Perpendicular lines meet to form
right angles.
7.
7. All right angles are congruent.
8.
8. Reflexive Property
9.
9. SAS
10.
10. CPCTC
11.
11. An isosceles triangle is a triangle
with two congruent sides.
12.
12. Contradiction steps 1 and 11
Practice online here:
http://regentsprep.org/Regents/mathb/1c/preprooftriangles.htm
http://regentsprep.org/Regents/mathb/1e/indirectpractice.htm
Activity:
http://regentsprep.org/Regents/mathb/1e/indirectteacher.htm
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Worksheets:
http://www.math.hawaii.edu/~woolcutt/2007_summer_100/homework_04.pdf
http://www.franklinroadacademy.com/teachers/zab/Geometry%28H%29/worksheets/Workshe
etTruthTableII.pdf
4. Linked Videos:
More proofs are presented in the following videos:
http://www.youtube.com/watch?v=vsluxs0B9Gg&feature=related
http://www.youtube.com/watch?v=zQ6Y0fismDg&feature=related
http://www.youtube.com/watch?v=b0SItpVQeqI