Writing proofs involves providing logical arguments to support statements using given information, definitions, axioms, theorems, and previously proven statements. There are two types of direct proofs: paragraph form and two-column proof. Paragraph form writes the proof as a paragraph, while two-column proof presents the statements in one column and reasons in another column. Indirect proof assumes the opposite of the statement is true, then arrives at a contradiction, proving the original statement must be true.

1. WRITING PROOFS

2. A proof is a logical argument in which each
statement is supported/justified by given
information, definitions, axioms, postulates,
theorems, and previously proven statements.
REMEMBER!
POSTULATE is a statement that is accepted to
be true without proof.
THEOREM is a statement accepted after is
proven deductively.

3. Proofs can be written of two kinds DIRECT and
INDIRECT proof.
DIRECT PROOF is a sequence of statements
which are either given or deductions from
previous statements, and whose last statement
is the conclusion to be proven.

4. 1. Paragraph Form
The Paragraph form is one way of proof where you
write a paragraph to explain why a conjecture for a
given situation is true.
Example
Given: ∠𝐻𝑂𝑃 𝑎𝑛𝑑 ∠𝑃𝑂𝐸 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
Prove: 𝐻𝑂⊥ 𝑂𝐸
Solution:
Since ∠𝐻𝑂𝑃 𝑎𝑛𝑑 ∠𝑃𝑂𝐸 are complementary, then 𝑚∠𝐻𝑂𝑃 + 𝑚 ∠𝑃𝑂𝐸
= 90 by definition of complementary angles. Thus, 𝑚∠𝐻𝑂𝑃 + 𝑚∠𝑃
𝑂𝐸=𝑚∠𝐻𝑂𝐸 by angle addition postulate and 𝑚∠𝐻𝑂𝐸=90 by
transitive property. So, ∠𝐻𝑂𝐸 is a right angle by definition of
right angles; therefore, 𝐻𝑂⊥ 𝑂𝐸̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ by definition of perpendicularity.

5. 2. Two-Column Proof
A two-column proof or the formal proof is proof presented in columns.
One for the statements and the other for reasons.
Example
Let us create a two-column proof of the example above.
Given: ∠𝐻𝑂𝑃 𝑎𝑛𝑑 ∠𝑃𝑂𝐸 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
Prove: 𝐻𝑂⊥ 𝑂𝐸
STATEMENTS REASONS
1. ∠𝐻𝑂𝑃 𝑎𝑛𝑑 ∠𝑃𝑂𝐸 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 1. Given
2. 𝑚∠𝐻𝑂𝑃+𝑚 ∠𝑃𝑂𝐸=90 2. Definition of complementary angles.
3. 𝑚∠𝐻𝑂𝑃+𝑚∠𝑃𝑂𝐸=𝑚∠𝐻𝑂𝐸 3. Angle addition postulate
4. 𝑚∠𝐻𝑂𝐸=90 4. Transitive Property
5. ∠𝐻𝑂𝐸 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 5. Definition of right angles
6. 𝐻𝑂⊥ 𝑂𝐸 6. Definition of perpendicularity

6. INDIRECT PROOF
An indirect proof is a method of reasoning
usually written in paragraph form. The
opposite of the statement to be proven is
assumed true until assumption leads to
contradiction.

7. Here are the easy steps to make an indirect proof.
1.Assume that the statement is false.
2.Proceed as you would in a direct proof.
3.Come across with contradiction
4.State that because of the contradiction,
it can’t be the case that the statement is
false, so it must be true.