# Ch 5: Triangles, Theorems and Proofs: Help and Review

### About This Chapter

## Who's it for?

Anyone who needs help understanding high school geometry material from will benefit from taking this course. You will be able to grasp the subject matter faster, retain critical knowledge longer and earn better grades. You're in the right place if you:

- Have fallen behind in understanding triangles, theorems and proofs.
- Need an efficient way to learn about triangles, theorems and proofs.
- Learn best with engaging auditory and visual tools.
- Struggle with learning disabilities or learning differences, including autism and ADHD.
- Experience difficulty understanding your teachers.
- Missed class time and need to catch up.
- Can't access extra math learning resources at school.

## How it works:

- Start at the beginning, or identify the topics that you need help with.
- Watch and learn from fun videos, reviewing as needed.
- Refer to the video transcripts to reinforce your learning.
- Test your understanding of each lesson with short quizzes.
- Submit questions to one of our instructors for personalized support if you need extra help.
- Verify you're ready by completing the Triangles, Theorems and Proofs chapter exam.

## Why it works:

**Study Efficiently:**Skip what you know, review what you don't.**Retain What You Learn:**Engaging animations and real-life examples make topics easy to grasp.**Be Ready on Test Day:**Use the Triangles, Theorems and Proofs chapter exam to be prepared.**Get Extra Support:**Ask our subject-matter experts any relevant question. They're here to help!**Study With Flexibility:**Watch videos on any web-ready device.

## Students will review:

In this chapter, you'll learn the answers to questions including:

- What are the guiding principles behind the ASA, SAS and SSS postulates?
- How do congruent proofs work?
- What is the converse of a statement?
- How do the AAS, HA and HL proofs work?
- What are the guiding principles behind the angle and perpendicular theorems?
- What procedures are used to prove the congruency of isosceles and right triangles?

### 1. Applications of Similar Triangles

Similar triangles are used to solve problems in everyday situations. Learn how to solve with similar triangles here, and then test your understanding with a quiz.

### 2. Triangle Congruence Postulates: SAS, ASA & SSS

When we have two triangles, how can we tell if they're congruent? They may look the same, but you can be certain by using one of several triangle congruence postulates, such as SSS, SAS or ASA.

### 3. Congruence Proofs: Corresponding Parts of Congruent Triangles

Congruent triangles have congruent sides and angles, and the sides and angles of one triangle correspond to their twins in the other. In this lesson, we'll try practice with some geometric proofs based around this theorem.

### 4. Converse of a Statement: Explanation and Example

Just because a conditional statement is true, is the converse of the statement always going to be true? In this lesson, we'll learn the truth about the converse of statements.

### 5. The AAS (Angle-Angle-Side) Theorem: Proof and Examples

When trying to find out if triangles are congruent, it's helpful to have as many tools as possible. In this lesson, we'll add to our congruence toolbox by learning about the AAS theorem, or angle-angle-side.

### 6. The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples

In this lesson, we'll learn about the hypotenuse angle theorem. With this theorem, we can prove two right triangles are congruent with just congruent hypotenuses and acute angles.

### 7. The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples

In this lesson, we'll learn about the hypotenuse leg theorem. This theorem enables us to prove two right triangles are congruent based on just two sides.

### 8. Perpendicular Bisector Theorem: Proof and Example

Perpendicular bisectors are multifunctional lines. They're not only perpendicular to the line in question, they also neatly divide it into two equal halves. In this lesson, we'll learn about the perpendicular bisector theorem.

### 9. Angle Bisector Theorem: Proof and Example

The angle bisector theorem sounds almost too good to be true. In this lesson, we set out to prove the theorem and then look at a few examples of how it's used.

### 10. Congruency of Right Triangles: Definition of LA and LL Theorems

In this lesson, we'll learn two theorems that help us prove when two right triangles are congruent to one another. The LA theorem, or leg-acute, and LL theorem, or leg-leg, are useful shortcuts for proving congruence.

### 11. Congruency of Isosceles Triangles: Proving the Theorem

Isosceles triangles have two equal sides. Are the base angles also equal? In this lesson, we'll prove how this is true. We'll also prove the theorem's converse.

### 12. Angle of Depression: Definition & Formula

This lesson will explore angles of depression with reference to angles of elevation. Using real world examples, we will seek to understand the similarities and differences of these angles.

### 13. Law of Syllogism in Geometry: Definition & Examples

This lesson will explain the law of syllogism and provide several examples showing when it can be used to reach a valid conclusion and when it cannot.

### 14. Midpoint Theorem: Definition & Application

Postulates and proven theorems allow us to know and understand certain things about geometric figures. In this lesson, we will be learning about the Midpoint Theorem.

### 15. Perpendicular Bisector: Definition, Theorem & Equation

How can we draw a triangle that will have two exactly equal length sides? Or what if we need to find the center of a circle that passes through a given set of points? In this lesson, we'll learn about the perpendicular bisector and how useful it can be in geometry!

### 16. Proof by Contradiction: Definition & Examples

One of several different ways to prove a statement in mathematics is proof by contradiction. Learn the definition of this method and observe how it is applied to proving a statement's truth value through examples and exploration.

### 17. Pythagorean Identities in Trigonometry: Definition & Examples

There are three trigonometry identities based off of the Pythagorean theorem. You will learn what they are in this lesson, as well as how to get from one to the other.

### 18. Square Matrix: Definition & Concept

A square matrix is a special type of matrix with an equal number of rows and columns. Learn more about square matrices in this lesson, including how to add and multiply them. Then, test your understanding with a short quiz.

### 19. Tetrahedral in Molecular Geometry: Definition, Structure & Examples

In this lesson, we'll learn what a tetrahedral is in molecular geometry. We'll also look at features and examples of tetrahedral structures. At the end of the lesson, take a brief quiz to see what you learned.

### 20. Proof by Induction: Steps & Examples

Mathematical induction is a method of proof that is often used in mathematics and logic. We will learn what mathematical induction is and what steps are involved in mathematical induction.

### 21. Ceva's Theorem: Applications & Examples

Ceva's theorem is an interesting theorem that has to do with triangles and their various parts. This lesson will state the theorem and discuss its application in both real-world and mathematical examples.

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### Other Chapters

Other chapters within the High School Geometry: Help and Review course

- Foundations of Geometry: Help and Review
- Logic in Mathematics: Help and Review
- Introduction to Geometric Figures: Help and Review
- Properties of Triangles: Help and Review
- Parallel Lines and Polygons: Help and Review
- Similar Polygons: Help and Review
- Quadrilaterals: Help and Review
- Circular Arcs and Circles: Help and Review
- Conic Sections: Help and Review
- Geometric Solids: Help and Review
- Analytical Geometry: Help and Review
- Introduction to Trigonometry: Help and Review