Geometry Theorems and Definitions Quiz

Mathematical Reasoning

• Reasoning is essential for problem-solving and decision-making in math, and is crucial for comprehending mathematical concepts and proofs. It is also valuable for practical problem-solving in daily life.

Inductive Reasoning

• Inductive reasoning involves formulating general statements based on specific observations.
• Conclusions are not guaranteed, but instead are likely based on patterns. This approach is based on observations to conclusions, it's a method of reasoning that makes generalizations.
• Examples of inductive arguments:
• Every quiz has been easy, so the test will be easy too.
• The teacher used PowerPoint in the past few classes, so the teacher will use it tomorrow.
• Every fall there have been hurricanes in the tropics, so there will be hurricanes this coming fall.

Deductive Reasoning

• Deductive reasoning starts with a general statement and draws specific conclusions.
• Conclusions are guaranteed if the initial statements are true. This approach is based on rules, facts, and established principles/theorems to reach a specific conclusion.
• In deductive reasoning, the statements must be true for the conclusion to be valid.
• Examples of deductive arguments:
• All students eat pizza. Claire is a student at ASU. Therefore, Claire eats pizza.
• All athletes work out in the gym. Barry Bonds is an athlete. Therefore, Barry Bonds works out in the gym.

Types of Reasoning & Examples

• Inductive:

• Observing patterns to conclude a general statement

• Analyzing specific examples to form a rule/principle

• Deductive:

• Starting with a general rule and using it to make specific conclusions

• Following logic and applying rules/principles to reach a conclusion

• The example of the 90% of humans being right-handed and Joe being a human is an example of deductive reasoning. The conclusion that Joe is right handed is logically guaranteed if the first two statements are true.

• Example identifying whether a triangle is isosceles involves understanding the concept of congruent sides and how to apply it to a specific figure.

• Example utilizing the "triangle sum property" and applying it to find an unknown angle is based on deductive reasoning. The property itself is a recognized theorem.

• Examples like the lion, and the catalog example show deductive reasoning.

Mathematical Systems

• Mathematical systems include undefined terms, definitions, postulates, and theorems:

• Undefined terms: Fundamental building blocks like point, line, and plane, which are not formally defined.

• Definitions: Clearly explain specific terms and their characteristics, showing the relationship between a defined term and undefined or other defined terms.

• Postulates (axioms): Basic assumptions which are accepted as true without proof.

• Theorems: Statements that can be proved using definitions and postulates.

• Corollaries are theorems that easily follow from other theorems. Lemmas are theorems that help prove later theorems because they are sub-proofs.

Geometry Symbols and Definitions

• Symbols are used to save time and space, and can also increase understanding of geometric shapes and relationships.
• Common geometric symbols used to represent triangle, congruent line segment, parallel lines etc.
• Definitions, such as midpoint and distance are important in geometric proofs.

Important Geometric Postulates (and theorems).

• Postulate 1 : Through any two points, there exists exactly one straight line.
• Postulate 2 : The measure of any line segment is a unique positive number.
• Postulate 3 : If X is a point on AB and A-X-B (X is between A and B), then AX + XB = AB.
• Postulate 4 : If two lines intersect, then they intersect in exactly one point.
• Postulate 5 : Through any three noncollinear points, there exists exactly one plane.
• Postulate 6 : If two planes intersect, then their intersection is a line.
• Postulate 7 : If two points lie in a plane then the line joining them lies in the same plane.
• Theorem 1.1: The midpoint of a line segment is unique

Test your understanding of important concepts related to geometry, including definitions, postulates, and theorems. This quiz covers key characteristics of definitions as well as foundational postulates and corollaries in geometry. Perfect for students looking to strengthen their knowledge in geometric principles.