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Worksheet 2.1
Introduction to Proof
1. Write each of the following statements in conditional form (“If…then…”).
a. Two angles are supplementary if their sum is a straight angle.
b. All numbers less than 6 are negative.
c. 3x3  10  14 when x  2 .
d. ( x  y)2 is positive whenever x is not equal to y .
2. Statements of Logic. As we have already discussed, a conditional statement is a statement
that can be written in if-then form.
For example:
If p, then q.
Other important logical statements (related to our given statement) are the…
Converse:
__________________________________
Inverse:
__________________________________
Contrapositive:
__________________________________
3. Write the converse of the following statement. Is it true?
If a ray bisects an angle, it divides the angle into two congruent angles.
4. Write the inverse of the following statement. Is it true?
If two angles are right angles, then the angles are congruent.
5. For the given statement, give the converse, the inverse, and the contrapositive. State whether
each of them is true or false, and provide a counterexample if it is false.
a. If an angle measures less than 90 , then it is an acute angle.
b. If Bertha does not live in Washington, then she does not live in Seattle.
6. Classify each of the following statements as true or false. If false, then briefly explain.
a. If p  4 , then p 2  16
b. If p 2  16 , then p  4
c. If p  4 , then p 2  16
d. If p 2  16 , then p  4
In order to present a proof that communicates clear and efficient use of mathematical principles,
we need to understand what we mean by principles in the context of geometry.
Mathematical principles are presented to us in various forms:
Definitions are mathematical statements that describe an object or idea. We aim definitions to be
precise, as they are used to make arguments concise and clear. Definitions are biconditional.
example – A quadrilateral is a four-sided figure.
Properties are mathematical statements based on algebraic truths/principles
 Reflexive Property:
A=A
o Used when one segment is part of two triangles, for example.
 Substitution:
If A = B, then A can be replaced by B in any expression.
 Addition/Subtraction:
If A = B, then A  C = B  C
 Multiplication:
If A = B, then A  C = B  C
Postulates/Axioms are mathematical statements assumed to be true
example – All radii of a circle are congruent.
Theorems are mathematical statements that can be proven. Note: theorems can only be used
once proven true.
example – The interior angles of a quadrilateral sum to 360 (we will prove this soon!).
7. Assumptions. When working with figures, it must be explicitly clear what we can and cannot
assume from a diagram. Consider the following diagram to help us arrive at assumptions we
can and cannot make:
What we CAN assume
What we CANNOT assume
8. Some Introductory Theorems
Below are three mathematical statements. Convince yourself that they are true.
If two angles are straight angles, then they are congruent.
If two angles are right angles, then they are congruent.
This one requires a few definitions first:
Supplementary angles: angles whose sum is 180 (or a straight angle/line).
Complementary angles: angles whose sum is 90 (or a right angle).
Bisect: to divide an angle/segment into two congruent parts.
If two angles are supplementary to congruent angles, then they are congruent.
C
P
S
D
B
H
A
T
Practice with Basic Proofs
Now that we have a few theorems and definitions to play with, we can try out or deductive
reasoning in more sophisticated arguments.
9. Example Proof: Support each statement with an appropriate reason.
B
Given: x  y
AF bisects BAC
CF bisects BCA
Prove: FAC  FCA
F
x
D
Statements
DE is a straight angle
1.
2.
BAC x  DAC  DE 
2.
3. BAC is supp. to x
BCA is supp. to y
3.
4. x  y
4.
5. BAC  BCA
5.
6. BAC  BAF  FAC
6.
7. BAF  FAC  BCF  FCA
7.
BCA  BCF  FCA
8.
AF bisects BAC ; CF bisects BCA
8.
9. BAF  FAC
9.
10. 2FAC  2FCA
10.
11. FAC  FCA
11.
BCF  FCA
C
Reasons
1.
BCA y  ECA  DE 
y
A
E
10. Another Example Proof
To produce a concise, yet thorough, argument to arrive at our conclusion, let’s consider
making a plan as to how we wish to proceed. Use this space below to make some notes for
yourself to use as a guide so that you can work efficiently:
Given:
J
GH  HJ
IH  HK
I
c
Prove: a  b
Statements
b
a
G
H
Reasons
K
11. Prove the following statement using a two column proof structure. Make sure to first
sketch a diagram using the labels of your choice. Then fill out the “Given” and “Prove”
information based on the diagram and the theorem stated below.
Theorem: If two angles are complementary to congruent angles, then they are congruent.
Diagram
Given:
Prove:
Statements
Reasons
12. Prove the following theorem:
Theorem: If two lines intersect, then opposite angles are congruent.
Given: AE intersects DB at C
A
D
C
Prove: ACB  DCE
E
B
Statements
Reasons