Download Lesson 4 - Novel Stars

Geometry
Chapter 4
Two-Column Proofs
Key Concepts:
By the end of this lesson you should understand:
¾ Conditional (if-then) statements
¾ Deduction
¾ Vertical Angle Theorem
In this lesson, we are going to learn about conditional statements, deductive reasoning
and two-column proofs. These tools are useful in more than just geometry. They also
will help you to develop logical thinking skills! So let’s see what we can learn.
Section 4-1
Section 4-1 gives us an introduction to conditional statements and deductive reasoning.
It is important, when writing a conditional, to identify the hypothesis and the conclusion
of the statement. After you have figured that part out, the rest is a snap! If you are not
quite sure what a conditional statement is, check out the Key Concepts.
Theorem 4-1
Point, Line, Plane Theorem
If a point is not on a line, then the point and the line determine exactly one
plane.
Theorem 4-2
Line Intersection Plane Theorem
If two distinct lines intersect, then they determine exactly one plane.
Problem:
Write a conditional statement for the following:
An isosceles triangle has two equal sides.
Solution:
A conditional describing this statement would be:
If a triangle is isosceles, then it has two equal sides.
p⇒q
The hypothesis (p) is, a triangle is isosceles.
The conclusion (q) is, it has two equal sides.
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Problem:
Indicate when the following conditional is false given the hypothesis (p) is true.
If Sue wears a jacket, then it is cold outside.
p⇒q
Solution:
This conditional would be false
if Sue wore a jacket, but it was warm outside.
This is false because the hypothesis (p): Sue wears a jacket, is true.
But, the conclusion (q): It is cold outside, is false.
Section 4-2
Section 4-2, gives us an introduction to two-column proofs. Two-column proofs are
really neat because you can see, step by step, how to get an answer. Let’s try to finish a
few two-column proofs. Once you get the hang of these, you will be able to write whole
proofs by yourself! But this is not the only way to write proofs. For more information or
extra help visit Dr. Math, one of the first places to go for any difficulties in math.
Problem:
You are given:
CT ≅ CA
And you are trying to prove that
∆CAT is isosceles.
C
Finish the following two-column proof.
T
STATEMENTS
1. CT ≅ CA
1. ?
2. CT = CA
2. ?
3. ∆CAT is isosceles.
3. ?
A
REASONS
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Solution:
STATEMENTS
1. CT ≅ CA
REASONS
1. Given (We say this is given
because this is one of the original pieces of
information to solve this problem.)
2. CT = CA
2. Definition of Congruent Segments (We
say this is by definition because the
definition of two congruent segments is that
they have the same length)
3. ∆CAT is isosceles.
3. Definition of an Isosceles Triangle (We
say this is by definition because the
definition of an isosceles triangle is that it
has two segments that are equal in length)
Section 4-3
Now we are given the chance to write two-column proofs by ourselves. Don’t worry if
you get stuck, just look at the other proofs done as examples and the summaries on how
to write a two-column proof and you should be fine. It is also good to remember that
theorems can be very useful in solving a proof. Let’s look at a few more examples.
Theorem 4-3
Theorem 4-4
Theorem 4-5
Theorem 4-6
Theorem 4-7
Congruent Supplements Theorem
If two angles are supplements of the same angle, then they are congruent.
Supplements of Congruent Angles Theorem
If two angles are supplements of congruent angles, then they are
congruent.
Congruent Complements Theorem
If two angles are complements of the same angle, then they are congruent.
Complement of Congruent Angles Theorem
If two angles are complements of congruent angles, then they are
congruent.
Vertical Angle Theorem
Vertical angles are congruent.
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Problem:
Given: ∠1 ≅ ∠2
Prove: ∠3 ≅ ∠4
Write a two-column proof.
Solution:
Statements
Reasons
1. ∠1 and ∠4 form a linear pair ;
∠2 and ∠3 form a linear pair
1. Definition of a linear pair
2. ∠1 and ∠4 are supplementary;
∠2 and ∠3 are supplementary
2. By the Linear Pair Postulate
3. ∠1 ≅ ∠2
3. Given
4. ∠3 ≅ ∠4
4. Theorem 4-4: If two angles are
supplements of congruent angles, then they
are congruent.
We are finished! What is really great about proofs like this is that we can see
how and why we got to each step of the proof. Proofs are meant to make math more
understandable, and two-column proofs make proofs more clear.
That’s it for lesson 4. Next time we are going to discover some stuff about perpendicular
lines and then we are going to learn polygons. Good luck on your submission. If you
have any trouble, don’t be afraid to review what we’ve learned. Have fun!
4
GLOSSARY
CONDITIONAL STATEMENTS
A conditional statement is a statement in if-then form, If p, then q, where p
and q are statements. In a conditional, p is the hypothesis and q is the
conclusion. These statements are usually written as p ⇒ q .
An example of a conditional statement is:
If Kelly is 18, then she is old enough to vote.
The hypothesis (p) is: Kelly is 18
The conclusion (q) is: She is old enough to vote
DEDUCTIVE REASONING
In deductive reasoning, conclusions are arrived at from accepted facts.
ISOSCELES
An isosceles triangle is a triangle with two congruent sides.
VERTICAL ANGLES
Vertical angles are the nonadjacent angles
formed when two lines intersect.
∠1 and ∠2 are vertical angles.
∠3 and ∠4 are vertical angles.
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1
4
2
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