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Geometry: Proofs





 



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A proof in geometry involves reaching a conclusion by following a sequence of statements that are logically connected.

It begins with a given statement which is called a hypothesis. Each statement is like a link in a chain in that it is related somehow to a preceding statement and/or a following statement. Each statement in geometry is supported by a reason for making that statement. Usually, the reason for the statement is given or is a rule used in geometry such as a postulate, definition or theorem.

A proof in geometry is usually written down in one of two formats.

The first is the two column table method with the statement written in the left column and the reason for the statement is written in the right column. The initial statement is written down first along with the reason which is listed as given. The next statements are listed so that they follow in the direction of the statement being proved.

With the flowchart method, the initial statement is written down inside a rectangular block. The reason for the statement is written down underneath the block that contains the statement. The statement that follows is written down in a block to the left or below the previous block along with the reason below it as before. An arrow is drawn pointing to the next block from the prior block. The last block is the statement that is being proven.

Drawing a diagram of the problem can be helpful in figuring out which statements should follow from the given statement. It can be used in any of the methods of proof.

There is some similarity between doing a proof in geometry to solving an algebra problem. The main difference is that when solving an algebra problem the rule that is used at each step is omitted when writing down the problem. The following example shows an algebra problem with the reason for each step listed to the right. This is similar to how a geometry proof is done.

Given 6x + 7 = 55, prove x = 8.

StatementReason
6x + 7 =55Given
6x = 48Subtraction Property
x = 8Division Property

The initial statement was given and the final statement was proven.

The next proof is an actual geometry proof:

Given:Intersecting lines l and m with angles angleA, angleB , angleC and angleD.
Prove:Vertical angles are equal in measure:

Intersecting lines create vertical angles that are equal.

Proof:

StatementReason
1. l, m intersect
angleA, angleC are vertical
angleB, angleD are vertical
Given
2. mangleA + mangleB = 180°
mangleB + mangleC = 180°
Supplementary angles are equal to 180 °
3. mangleA + mangleB = mangleB + mangleC Substitution
4. mangleA = mangleCSubtraction

In step 3, the sum of the measures are both equal to 180° and therefore would be equal to each other. This allows one sum to be substituted for the value of 180°.

In step 4, the measures of the vertical angles angleA and angleC are proven to be equal. The proof just given was done using the two column table. It can also be done with a flowchart.

A flowchart can be used as a geometry proof.

Since the two column table is the most frequently used method of doing geometry proofs it will be the primary method of showing how to do proofs in the following examples.

The next example deals with proving that the sum of the measures of two angles are equal to the measure of another angle using angle bisectors. The bisector of an angle splits the angle into two angles with equal measure. The proof also demonstrates the use of the Angle Addition Postulate.

The sum of angle measures created by angle bisectors is equal to the sum of the other angle measures created by the same angle bisectors.
Given:angleBAC, angleCAD, angleDAE, angleEAF, Ray AC bisects angleBAD, Ray AE bisects angleDAF.
Prove:
mangleBAC, + mangleEAF = mangleCAE

Proof:

StatementReason
1. Ray AC bisects angleBAD
Ray AE bisects angleDAF
1. Given
2. mangleDAE = mangleEAF
mangleBAC = mangleCAD
2. Bisected angles are equal in measure.
3. mangleBAC + mangleEAF = mangleCAD + mangleDAE 3. Addition
4. mangleCAD + mangleDAE = mangleCAE4. Angle Addition
5. mangleBAC + mangleEAF = mangleCAE5. Substitution

Segment Addition Postulate

The next example deals with the result of adding line segments.

Given 4 points on a line segment with two of the points as endpoints prove that the smaller line segments add up to the larger one.

The Segment Addition Postulate says that the lengths of the pieces of a line segment add up to the total length of that line segment.
Given:A-B-C-D on
AD
Prove:
AB + BC + CD = AD

Proof:

StatementReason
1. Given:
A-B-C-D on
AD
1. Given
2. AB + BD = AD2. Segment Addition
3. BC + CD = BD3. Segment Addition
4. AB + BC + CD = AD4. Substitution

Some of the reasons given in a geometry proof can be repeated or two adjacent steps that give the same reason can be combined into a single step.

 




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