Back To Course

Geometry: High School15 chapters | 160 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

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

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
2 in chapter 1 of the course:

Back To Course

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
- Go to High School Geometry: Foundations of Geometry

- Computer Science 109: Introduction to Programming
- Introduction to HTML & CSS
- Introduction to JavaScript
- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Early Civilizations & The Ancient Near East
- Fundamental Overview of World War I
- The Virginia Dynasty & Jacksonian America
- 1920's America and the Great Depression
- Building the United States After the American Revolution
- CEOE Test Cost
- PHR Exam Registration Information
- Claiming a Tax Deduction for Your Study.com Teacher Edition
- What is the PHR Exam?
- Anti-Bullying Survey Finds Teachers Lack the Support They Need
- What is the ASCP Exam?
- ASCPI vs ASCP

- Subtraction in Java: Method, Code & Examples
- Hydrogen Chloride vs. Hydrochloric Acid
- Extraction of Aluminum, Copper, Zinc & Iron
- Iroquois Culture, Traditions & Facts
- Noun Clauses Lesson Plan
- Adverb of Manner Lesson Plan
- Timeline Project Ideas for High School
- Quiz & Worksheet - Multi-Dimensional Arrays in C
- Quiz & Worksheet - What is a Diastereoisomer?
- Quiz & Worksheet - Mauryan Empire Art & Culture
- Quiz & Worksheet - What is a Convergent Sequence?
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Bullying in Schools | Types & Effects of Bullying
- Parts of Speech Worksheets

- NY Regents Exam - Chemistry: Help and Review
- Intro to Sociology Syllabus Resource & Lesson Plans
- GACE Middle Grades Social Science (015): Practice & Study Guide
- Earth Science: High School
- Technical Writing Syllabus Resource & Lesson Plans
- Saxon Algebra 1/2: Whole Numbers & Operations With Whole Numbers
- Scientific Inquiry - ORELA Middle Grades General Science
- Quiz & Worksheet - Characteristics of the English Enlightenment
- Quiz & Worksheet - Types of Writing for Self-Expression
- Quiz & Worksheet - Social Process Theory
- Quiz & Worksheet - Volume & Surface Area of a Torus
- Quiz & Worksheet - Domestic Terrorism History & Types

- The Arab-Islamic Empire: Emergence, Establishment & Expansion
- William Penn: History, Facts & Biography
- Free Online Accounting Courses with a Certificate
- Ohio Alternative Teacher Certification
- Illinois Common Core Social Studies Standards
- Holocaust Lesson Plan
- How to Learn French
- Creative Writing Exercises for Middle School
- Fairfax County Adult Education
- What does Redshirt Mean in College Sports?
- NATA Certification Requirements
- Common Core Standards in Delaware

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject