 # Reasoning in Geometry

These lessons are part of a series of free, online High School Geometry Lessons.

We also have more videos, worksheets, and activities to help Geometry students.

In these lessons, we will learn

• Inductive reasoning
• Deductive reasoning

### Inductive Reasoning

Inductive reasoning is the process of observing, recognizing patterns and making conjectures about the observed patterns. Inductive reasoning is used commonly outside of the Geometry classroom; for example, if you touch a hot pan and burn yourself, you realize that touching another hot pan would produce a similar (undesired) effect.

The conclusion you draw from inductive reasoning is called the conjecture. A conjecture is not supported by truth.

When making a conjecture, it is possible to make a statement that is not always true. Any statement that disproves a conjecture is a counterexample.

Examples:
1. Determine the number of points in the 4th, 5th, and 8th figure.

2. a) Determine the next 2 terms of the sequence.
4,8,16,32,64, ...
b) Determine a formula that could be used to determine any term in the sequence. This video will define inductive reasoning, use inductive reasoning to make conjectures, determine counterexamples.

How to define inductive reasoning, how to find numbers in a sequence?
Inductive Reasoning & Conjectures
Use inductive reasoning to identify patterns and make conjectures.
Example:
Describe the pattern in the numbers
-7, -21, -63, -189, ...
And write the next three terms in the series.
Proving Conjectures & Counterexamples
To show that a conjecture is true is tricky, we have to show that the conjecture is true for ALL cases.
To show that a conjecture is false is comparatively easier. We just have to show that the conjecture doesn't hold for ONE case. This case is a counterexample.
How to find counterexamples to disprove conjectures?
Example:
Provide a counterexample for the following conjectures.
(a) All prime numbers are odd.
(b) (a + b)2 = a2 + b2
(c) If the product of two numbers is positive, than the two numbers must both be positive.

### Deductive Reasoning

Deductive reasoning is the process of reasoning logically from given statements to make a conclusion. Deductive reasoning is the type of reasoning used when making a Geometric proof, when attorneys present a case, or any time you try and convince someone using facts and arguments.

How to define deductive reasoning and compare it to inductive reasoning?
Example:
1. Prove QUAD is a parallelogram.
2. Draw the next shape.
This video defines deductive reasoning and the basic rules of logic
Deductive reasoning is when you make conclusions based upon facts that support the conclusion without question.
Law of Detachment
Law of Contrapositive
Law of Syllogism

Difference between inductive and deductive reasoning
Example:
To estimate the population of a town in upcoming years, one of the town workers collected population from past years and made this table:
The town wants to estimate the population for 2015, 2018, and 2020. To do this, will you be using inductive reasoning or deductive reasoning?
Deductive Reasoning Problem: Sequence, series and induction
Example:
Hiram solved the equation using the following steps
Is this an example of deductive reasoning?
Deductive Reasoning Problem
Example:
Use deductive reasoning and the distributive property yo justify (x + y)2 = x2 + 2xy = y2. Provide the reasoning for each step.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.  