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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson, and you will learn how important inductive and deductive reasoning is in the field of mathematics, especially when dealing with proofs in geometry. Learn how these two fundamental forms of reasoning give rise to formal theorems.

**Inductive** and **deductive** reasoning are two fundamental forms of reasoning for mathematicians. The formal theorems and proofs that we rely on today all began with these two types of reasoning. Even today, mathematicians are actively using these two types of reasoning to discover new mathematical theorems and proofs. Believe it or not, you yourself might be using inductive and deductive reasoning when you make assumptions about how the world works.

Defined, **inductive reasoning** is reaching a conclusion based off of a series of observations. A conclusion that is reached by inductive reasoning may or may not be valid. An example of inductive reasoning is, for example, when you notice that all the mice you see around you are brown and so you make the conclusion that all mice in the world are brown. Can you say for certain that this conclusion is correct? No, because it is based on just a few observations. However, this is the beginning of forming a correct conclusion, or a correct proof. What this observation has given you is a starting hypothesis to test out.

Inductive reasoning typically leads to **deductive reasoning**, the process of reaching conclusions based on previously known facts. The conclusions reached by this type of reasoning are valid and can be relied on. For example, you know for a fact that all pennies are copper colored. Now, if your friend gave you a penny, what can you conclude about the penny? You can conclude that the penny will be copper colored. You can say this for certain because your statement is based on facts.

So, how does inductive and deductive reasoning figure into geometry? Well, inductive reasoning is the beginning point of proofs, as it gives you a hypothesis you can test out, similar to what we discussed with the mice. For example, we could observe that all three angles of several pairs of triangles are equal and that each pair of triangles look the same, except that one is bigger than the other. Through inductive reasoning, we can reach the conclusion that if two triangles have angles that all measure the same, then they are similar triangles.

But is this reliable? Not yet, because it is not based on facts. However, it does become our hypothesis that we can test out in order to make a correct and valid conclusion. We can use deductive reasoning now to begin making correct conclusions. We look for facts that we know. What do we know? We know for a fact that there is a formal theorem that has been proved time and time again that tells us that if two triangles have the same angles, then they are similar.

If we know this, and we know that the two triangles we are looking at do indeed have the same angles, then we can say for certain that the two triangles are similar. Because our conclusion is based on facts, the conclusions reached by deductive reasoning are correct and valid. Simply put, inductive reasoning is used to form hypotheses, while deductive reasoning is used more extensively in geometry to prove ideas.

What have we learned? We've learned that **inductive reasoning** is reasoning based on a set of observations, while **deductive reasoning** is reasoning based on facts. Both are fundamental ways of reasoning in the world of mathematics. All of the formal theorems and proofs started out with one mathematician making a hypothesis based on inductive reasoning from what he or she observed. After this initial observation, the mathematician switched to deductive reasoning to prove that what he or she observed is indeed true and based on facts.

Inductive reasoning, because it is based on pure observation, cannot be relied on to produce correct conclusions. Deductive reasoning, on the other hand, because it is based on facts, can be relied on. Because the world of math is all about facts, deductive reasoning is relied on instead of inductive reasoning to produce correct conclusions. Inductive reasoning is relied on to produce hypotheses and new ideas that can be tested and proved using other more reliable methods.

Study this video's information so that you may have the ability to:

- Distinguish between inductive and deductive reasoning
- Showcase knowledge of the use of both types of reasoning in mathematics

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Geometry: High School15 chapters | 160 lessons

- What is Geometry? 4:36
- Inductive & Deductive Reasoning in Geometry: Definition & Uses 4:59
- The Axiomatic System: Definition & Properties 5:17
- Euclid's Axiomatic Geometry: Developments & Postulates 5:58
- Undefined Terms of Geometry: Concepts & Significance 5:23
- Properties and Postulates of Geometric Figures 4:53
- Algebraic Laws and Geometric Postulates 5:37
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