THE KEYS TO WRITING GEOMETRY PROOFS

January 6, 2025

Every geometry teacher has heard it. “Proofs are so HAAAARRRRDDD.” (If you have any experience with teenagers, you know how to pronounce that last word.) I’m going to give you some hints to let you know that they don’t have to be as hard as you think.

First. Remember that there are two different ways to write a proof – Direct and Indirect.

Direct proofs have three methods – Two Column, Flowchart, and Paragraph.

An Indirect Proof is also called a Proof by Contradiction.

Direct Proofs

For direct proofs, it doesn’t matter which format you use, you’re still going to use the same logic, you’re just presenting it in a different visuality. You are giving your statement and providing an explanation (reason) for that statement to logically be justified from the previous statements.

The first statement is easy. Repeat what you’ve been given.

The second statement is going to use something from the given information to logically build towards what you are trying to prove. For example, if you are told that the measures of two angles are equal, then the second statement may be that the angles are congruent. The reason is the definition of congruency.

From there, you need to work on the proof towards your end goal.

Key points to remember:

If you can, prove that two triangles within the figure are congruent and then you can use CPCTC. CPCTC is your friend!
Know your definitions.
- Definition of congruency
- Definition of complementary
- Definition of supplementary
- Definition of parallel lines
  - Corresponding angles are congruent
  - Alternate exterior angles are congruent
  - Alternate interior angles are congruent
  - Same side interior angles are supplementary
- Definition of perpendicular lines – especially that perpendicular lines create right angles.
Important Theorems and Postulates
- Addition Property of Equality
- Subtraction Property of Equality
- Segment Addition Postulate
- Angle Addition Postulate
- Midpoint Theorem
- Vertical Angles Theorem

Example of a Two-Column Proof

Given: ∠BCD is a right angle. ∠1 ≌ ∠3

Prove: ∠1 and ∠2 are complementary.

Example of Flowchart Proof

Given: ∠A and ∠C are complementary

∠B and ∠C are complementary

Prove: ∠A ≌ ∠B

When you look at this question, it is intuitive to think “of course, if A and B are both complementary to C, then they must be congruent.” However, we don’t know why. We have to write our series of steps and justifications. This time we will show it in a flow chart.

Example of Paragraph Proof

A paragraph proof is just like a two-column proof or a flowchart proof. The difference is that you will write each statement and justification in a sentence.

Given: Parallelogram ABCD

Prove: AE ≌ EC and ED ≌ BE

We are given parallelogram ABCD. By definition of a parallelogram, AB || DC and AD || BC.

Since AB || DC, ∠BAC ≌ ∠DCA and ∠ABD ≌ ∠BDC, because they are alternate interior angles of the parallel lines.

Likewise, ∠DAC ≌ ∠BCA and ∠ADB ≌ ∠DBC, because AD || BC.

Since opposite sides of a parallelogram are congruent, AB ≌ DC and AD ≌ BC.

We can now say that ∆AED ≌ ∆CEB and ∆AEB ≌ ∆CED by reason of ASA.

Therefore, because of CPCTC, AE ≌ EC and ED ≌ BE.

Indirect Proofs

One way to prove that a statement is true is to temporarily assume that what you are trying to prove is false. By showing this assumption to be logically impossible, you prove your assumption to be false and the original conclusion true. This is known as an indirect proof.

An indirect proof is written as a paragraph.

Steps for writing an indirect proof

Assume that the conclusion is false by assuming the opposite is true.
Use if-then statements to show that this assumption leads to a contradiction of the hypothesis or some other fact.
Point out that the assumption must be false, and therefore, the conclusion must be true.

Example of an Indirect Proof

Given: ∆ABC, m ∠C = 100

Prove: ∠A is not a right angle

Step 1 Assume that ∠A is a right angle

Step 2 Show that this leads to a contradiction.

If ∠A is a right angle, then m ∠A = 90.
If m ∠A = 90, then m ∠C + m ∠A = 100 + 90 = 190.
Thus, the sum of the measures of the angles of ∆ABC is greater than 180.

Step 3 The conclusion that the sum of the measures of the angles of ∆ABC is greater than 180 is a contradiction of the known property – the sum of the three angles of a triangle is equal to 180^o. The assumption that ∠A is a right angle must be false, which means that the statement “∠A is not a right angle” must be true.

PRACTICE IT

What would be the assumption statement (step 1) for each of these proofs if you were going to write an indirect proof.

Given: ∆XYZ, m ∠X = 100

Prove: ∠Y is an acute angle

Given: Transversal t cuts lines a and b, m∠1 ≠ m∠2

Prove: a is not parallel to b

For questions 3 and 4 you will use this figure:

Given: ∠1 ≅ ∠2, OJ is not ≅ to OK

Prove: ∠J and ∠K are not both right angles.

Given: OJ ≅ OK; JE is not ≅ to KE

Prove: OE doesn’t bisect ∠JOK

I hope these tips and keys help to unlock proofs for you. Keep on practicing and soon they won’t seem quite as hard. You can also contact me, Mary Lou, to meet and practice together. Click here to sign up for one-on-one practice time.

Here are some links to pages with more practice: