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High School Geometry: Homework Help Resource13 chapters | 142 lessons

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Lesson Transcript

Instructor:
*Mia Primas*

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

Have you ever made a statement that someone challenged you to prove to be true? You have to explain things in a logical and indisputable way in order to do that. In this lesson, you'll learn different ways that statements can be proven in mathematics.

Writing a mathematical proof is similar to an attorney arguing a case in a courtroom. An attorney's task is to prove a person's guilt or innocence using evidence and logical reasoning. A **mathematical proof** shows a statement to be true using definitions, theorems, and postulates. Just as with a court case, no assumptions can be made in a mathematical proof. Every step in the logical sequence must be proven. Mathematical proofs use **deductive reasoning**, where a conclusion is drawn from multiple premises. The premises in the proof are called statements.

Proofs can be direct or indirect. In a **direct proof**, the statements are used to prove that the conclusion is true. An **indirect proof**, on the other hand, is a proof by contradiction. It begins by assuming the opposite of the statement that is to be proven. During the proof, a contradiction will be reached, showing that the assumed statement is false. For the examples in this lesson, we will use direct proofs since they are used more commonly.

The format of a proof can be a simple paragraph, a flow chart, or a two-column chart. We will look at an example of each.

All proofs should begin with the given information that is provided. When writing a paragraph proof, each sentence provides a statement and explanation leading to the conclusion. The conclusion is the statement that is being proven.

For example:

- Given that angle
*AED*is a right angle, prove that angle*AEC*measures ninety degrees.

We are given that angle *AED* is a right angle. According to the definition of right angles, it's therefore ninety degrees. Angle *AED* and angle *AEC* are linear and supplementary, based on the definition of linear angles. If they are supplementary, then they have a sum of 180 degrees. Therefore, since *AED* is ninety degrees, angle *AEC* must also be ninety degrees.

Next, we'll see how this proof can be made using a flow chart.

A flow chart proof uses a diagram to show each statement leading to the conclusion. Arrows are drawn to represent the sequence of the proof. The layout of the diagram is not important, but the arrows should clearly show how one statement leads to the next. The explanation for each statement is written below the statement:

A two-column proof contains a left column where each statement is listed in a separate row and a right column with the explanation, or reason, for each statement. The statements and reasons are often numbered to clearly show how they are paired:

Statement | Reason |
---|---|

1. Angle AED is a right angle |
1. Given |

2. Angle AED = 90 degrees |
2. Definition of right angle |

3. Angle AED and angle AEC are linear and supplementary |
3. Linear angle theorem |

4. Angle AED + Angle AEC = 180 |
4. Definition of supplementary |

5. 90 + Angle AEC = 180 |
5. Substitution property (2) |

6. Angle AEC = 180 - 90 = 90 |
6. Subtraction property of equality |

Notice that reason number five references statement two. Doing this explains that angle *AED* is substituted with ninety degrees because they are equivalent.

Mathematical proofs use deductive reasoning to show that a statement is true. The proof begins with the given information and follows with a sequence of statements leading to the conclusion. Each statement is supported with a definition, theorem, or postulate. Proofs can be formatted as a paragraph, flow chart, or a two-column chart.

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High School Geometry: Homework Help Resource13 chapters | 142 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Converse of a Statement: Explanation and Example 5:09
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
- The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples 5:50
- The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples 6:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
- Mathematical Proof: Definition & Examples 3:41
- Side-Angle-Side (SAS) Triangle: Definition, Theorem & Formula 4:57
- Square Prism: Definition & Examples 3:04
- Triangle Inequality: Theorem & Proofs
- Two-Column Proof in Geometry: Definition & Examples 5:48
- What is a Vector in Math? - Definition & Examples 6:27
- What is an Acute Angle? - Definition, Description & Examples 1:45
- Go to Triangles, Theorems and Proofs: Homework Help

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