Chapter 1 Section 4Beginning Proofs

With Calvin and HobbesCaroline Andrews

Geometry~8May 30, 2008

Beginning Proofs A two-column geometric proof consists of a list of

statements, and the reasons that we know those statements are true. The statements are listed in a column on the left, and the reasons for which the statements can be made are listed in the right column. Every step of the proof (that is, every conclusion that is made) is a row in the two-column proof, and the reason is next to it.

You should always number your statements and their corresponding reasons to prevent confusion and make your proofs more organized.

Introduction A theorem is a mathematical statement that can

be proved. It is usually and “If-then” statement.

Theorem 1- If two angles are right angles, then they are congruent.

Theorem 2-If two angles are straight angles, then they are congruent.

Creating Proofs This is an example of the format of a

two-column proof: Statement Reason______________________________1. 1. 2. 2. 3. 3.

When asked to write a two-column proof, you will be given a diagram and a number of statements to assist you in your proof. These will be listed as givens. And you should list them in your proof under statements. Then, in the reason column, you can specify, “Given”.

Additional Information Because of the definition of congruent angles, if 2 angles

have the same measure, they are congruent. Because of the definition of congruent segments, if 2

segments have the same measure, they are congruent. Because of the definition of a right angle, if and angle has

the measure of 90, it is a right angle. Because of the definition of a straight angle, if an angle has the measure of 180, then it is a straight angle. All of the above definitions can be used as reasons for statements in your proofs.

ProofsUsing the theorem 1 and the format of a proof, we can now create a proof proving two right anglesLet’s see an example problem.

Sample Problem #1 1. A is a right 1. Given

2. B is a right 2. GivenGiven: A is a right 3. A B 3. If two s are

B is a right right s, then Prove: A B they are .

A B

C D

ProofsNow let’s see a proof involving theorem 2.

Sample Problem #2 1. Diagram as shown 1. Given 2. EFG is a straight 2. Assumed from diagram

Given: Diagram as shown EFG HFJ 3. HFJ is a straight 3. Same as 2

4. EFG HFJ 4. If 2 s are straight s, then they are .

As you can see, sometimes the given will be simply in the form of the diagram, since it is always

safe to assume that an angle is a straight angle when it looks like one in the diagram. Also, when a reason is repeated twice, you can shorten the amount of writing you have to do by stating that the reason is the same as a previous step, as shown in step 3 of the above proof.

Proofs Sample Problem #3 1. A = 90 1. Given 2. ACB = 40 2. GivenGiven: A = 90 3. ACD = 50 3. Given

ADB = 40 4. A is a right 4. If an 's measure is

90 BDC = 50 then it is a right .Prove: ADC A 5. ADC = 90 5. AdditionA B 6. ADC is a right 6. Same as 4 7. ADC A 7. If two s are right

s, then they are .

D C

Practice Problems Pg. 26-27 #3-5,7

Answers to Practice Problems

#3. Statements1. A is a right 2. B is a right 3. A BReasons1. Given2. Given3. If 2 s are right s then they are .

More Answers #4Statements1. CDE = 1102. FGH = 110 3. CDE FGHReasons1. Given2. Given3. If 2 s have the same measure then they are .

More Answers #5Statements1. JK = 2.5 cm2. NO = 2.5 cm3. JK NOReasons1. Given2. Given3. If 2 segments have the same measure then they

are .

More Answers #7Statements Reasons1. 1 = 20 1. Given2. 2 = 40 2. Given3. 3 = 30 3. Given4. XYZ = 90 4. Addition5. XYZ is a right angle 5. If an ‘s measure is

90, then it is a

right .

Works Cited"Math a Formula." Math A. 1991. Oswego City School Districts.

27 May 2008 <http://regentsprep.org/Regents/math/forsheet/Formula.htm>.

Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. New Edition ed. Evanston, Illinois: McDougal Littel, 1991. 23-27.

The End