About This Chapter
Introduction to Proofs and Constructions - Chapter Summary
If you're looking to get a better idea of the differences between proofs, theorems and postulates, this chapter is for you. Introductory lessons explain what theorems are used for and introduce you to the axiomatic systems and postulates used to derive them.
You can build on this knowledge to explore theorems relating to corresponding pairs of angles and sides as well as parallel or perpendicular lines. Instructors also walk you through the various postulates used to demonstrate the congruence of triangles or establish the relationships between triangle medians, midsegments and centroids.
You can also take a look at parallelogram theorems used to prove the congruence of opposite sides and angles, the supplementary nature of consecutive angles and the shapes formed by their diagonals. This chapter concludes with lessons that show you how to apply this knowledge by using compasses and rulers to create geometric shapes, such as equilateral triangles, squares, triangles and inscribed hexagons. Topics of instruction include:
- Axiomatic systems
- Triangle congruence postulates
- Postulates for lines and angles
- Triangle theorems
- Parallelogram theorems
Whether you choose to access lesson content in the form of short videos or corresponding transcripts, you'll get the benefit of our instructors' expertise. This chapter also includes self-checking, multiple-choice quizzes you can take as many times as you need.
1. The Axiomatic System: Definition & Properties
Learn what kinds of things are included in an axiomatic system in this video lesson. Also learn why consistency, independence, and completeness are important in axiomatic systems.
2. Postulates & Theorems in Math: Definition & Applications
In this lesson, we will define postulates and theorems in mathematics. We will look at examples of postulates and theorems and how to use them in mathematics and in real-world applications.
3. Interior Angle Theorem: Definition & Formula
This lesson will define what an interior angle is, and it will provide and explain how to use the formula for finding the sum of the interior angles of a polygon. Examples will be provided detailing four of the ways it can be used.
4. Alternate Interior Angles: Definition, Theorem & Examples
In this lesson, you will learn how to identify alternate interior angles and how to use the theorem to find missing angles and to solve everyday geometry problems.
5. Supplementary Angle: Definition & Theorem
There are many important angle pairs in geometry. In this lesson, we'll learn about supplementary angles, including what they are, where they are used, and why you need to know about them.
6. Corresponding Angles: Definition, Theorem & Examples
In this lesson, you will learn how to identify corresponding angles. You will also learn how to use a theorem to find missing angles and solve everyday geometry problems.
7. Geometric Proofs: Definition and Format
Do you have something to prove? Can you explain why? In this lesson, we'll learn all about geometric proofs, including the parts that comprise a proof.
8. Methods & Tools for Making Geometric Constructions
Did you know that it's possible to do math without using numbers? This is exactly what Euclid did when he showed how to solve mathematical problems by drawing them out instead of with numbers.
9. Practice Making Geometric Constructions with Tools
There are a number of tools available to geometry students to help make their work a bit easier. In this lesson, we will review how to use many of them in order for you to get the most out of every geometry class.
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Other chapters within the TExES Mathematics 7-12 (235): Practice & Study Guide course
- About the TExES Math 7-12 Exam
- Real Numbers
- Mathematical Models
- Complex Numbers & the Complex Plane
- Number Theory
- Number Patterns
- Functions and Graphs
- Linear Functions
- Quadratic Functions & Polynomials
- Evaluating Piecewise & Composite Functions
- Rational and Radical Functions
- Inequalities and Absolute Values
- Exponentials & Logs
- The Unit Circle
- Trigonometric Functions
- Using a Scientific Calculator for Calculus
- Understanding Limits in Math
- Understanding Rate of Change
- Calculating Derivatives of Functions
- Derivatives and Graphs
- Optimization in Calculus
- Definite Integrals and Sums
- Integration Applications in Calculus
- Working with Measurement
- Finding Volume, Area & Perimeter
- Congruence and Similarity
- Real World Shapes
- Coordinate Geometry
- Understanding Transformations in Math
- Conic Sections
- Understanding Vectors
- Measuring & Displaying Data
- Data Distribution Overview
- Sampling in Statistics
- Distribution & Inference in Statistics
- Inference About a Mean
- Regression and Correlation
- Finding Probability
- Probability Distributions and Statistical Inference
- Experiments and Surveys
- Mathematical Process & Perspectives
- Teaching Strategies & Activities for the Math Classroom
- Differentiated Instructional Strategies for the Math Classroom
- Using Student Assessments in the Math Classroom
- TExES Mathematics 7-12 Flashcards