Writing Proofs

1. Math 8 – Proofing (Direct
and Indirect)
Ms. Andi Fullido
© Quipper

2. Objectives
• At the end of this lesson, you should be able to:
• differentiate between a direct and indirect proof;
• use two forms of representing proofs;
• write a direct proof using paragraph or two-column form; and
• write an indirect proof using paragraph or two-column form.

3. Vocabs
• A proof is an organized set of statements and reasons to
establish the truth of a specific statement. The properties of real
numbers may be combined with some definition and postulates
to prove the validity of statements.
• A proof may be presented in different ways. The most popular
are the paragraph form and the two-column form.
• In a two-column proof, logical statements are written on the
left side and its corresponding reasons on the right side. It often
uses symbols of different relationships.

4. Vocabs
• In a paragraph proof, the logical statements are presented
using complete sentences backed up by reason.
• In both methods, the deductive reasoning approach is used.
The proof begins with given information and ends with the prove
statement.

5. Vocabs
•≅ - approximately equal to or congruent
to
•∠AB – angle AB
•𝑀𝐵- Line MB

6. Direct Proof
• Suppose you are given a premise p and you want
to prove that a conclusion q is true.
• The direct proof would assume that p is true,
then use, in the context of geometry, properties,
postulates, definitions and theorems to show
that q is true.

7. Writing Direct Proofs
1. State the given. These statements are
considered facts, therefore, true.
2. State what to prove.
3. Draw a figure which can serve as a guide in
establishing the proof.
4. Present the proof using a preferred method
(two-column or paragraph).

8. Given: M is the midpoint of 𝐴𝐵
Prove: 𝐴𝑀 ≅ 𝐵𝑀

9. Given: M is the midpoint of 𝐴𝐵
Prove: 𝐴𝑀 ≅ 𝐵𝑀
• Proof in paragraph form:
• Given that point M is the midpoint of line
segment AB, by the definition of a midpoint, the
measurement of AM and BM are equal.
Since 𝐴𝑀 = 𝐵𝑀, then by definition of congruent
segments, we can say that line segment AM is
congruent to line segment BM, or 𝐴𝑀 ≅ 𝐵𝑀.

10. Indirect Proof
• Given a premise p and a conclusion q,
an indirect proof would assume that q is false.
You would then use the same properties,
postulates, definitions and theorems to show
that p would also be false by arriving at a
contradiction.

11. Writing Indirect Proofs
• Accept the given statement is true.
• Assume the opposite of the statement to be proved.
• State the reasons directly until there is a contradiction of the
given or the other statements.
• State that the assumption of the opposite of the statement to be
proved must be false.
• Follow Steps 3 and 4 of writing direct proofs.

12. Given: M is not the midpoint of 𝐴𝐵
Prove: 𝐴𝑀 ≠ 𝑀𝐵

13. Given: M is not the midpoint of 𝐴𝐵
Prove: 𝐴𝑀 ≠ 𝑀𝐵
• Proof in two-column form: