# Ch 5: Triangles, Theorems and Proofs: Tutoring Solution

### About This Chapter

## How it works:

- Begin your assignment or other high school geometry work.
- Identify the triangles, theorems and proofs concepts that you're stuck on.
- Find fun videos on the topics you need to understand.
- Press play, watch and learn!
- Complete the quizzes to test your understanding.
- As needed, submit a question to one of our instructors for personalized support.

## Who's it for?

This chapter of our high school geometry tutoring solution will benefit any student who is trying to learn about triangles, theorems and proofs and earn better grades. This resource can help students including those who:

- Struggle with understanding triangle congruence postulates, congruence proofs, similarity transformations or any other triangles, theorems and proofs topic
- Have limited time for studying
- Want a cost effective way to supplement their math learning
- Prefer learning math visually
- Find themselves failing or close to failing their triangles, theorems and proofs unit
- Cope with ADD or ADHD
- Want to get ahead in high school geometry
- Don't have access to their math teacher outside of class

## Why it works:

**Engaging Tutors:**We make learning about triangles, theorems and proofs simple and fun.**Cost Efficient:**For less than 20% of the cost of a private tutor, you'll have unlimited access 24/7.**Consistent High Quality:**Unlike a live algebra tutor, these video lessons are thoroughly reviewed.**Convenient:**Imagine a tutor as portable as your laptop, tablet or smartphone. Learn about triangles, theorems and proofs on the go!**Learn at Your Pace:**You can pause and rewatch lessons as often as you'd like, until you master the material.

## Learning objectives

- Solve problems with similar triangles.
- Use SAS, SSS and ASA triangle congruence postulates.
- Utilize the CPCTC theorem.
- Provide an example of the converse of a statement.
- Explore the use of similarity transformations.
- Prove the relationships of geometric figures, and solve practice problems.
- Provide proof of the AAS, SA and HL theorems.
- Explore the use of the perpendicular bisector theorem and the angle bisector theorem.
- Define the LL and LA theorems.
- Explain the congruency of the base angles of isosceles 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 Elevation: Definition, Formula & Examples

This lesson will define an angle of elevation, and it will provide some basic skills necessary to calculate the measure of one of these types of angles or to use it to calculate another value. Two example problems are also provided.

### 13. Centroid: Definition, Theorem & Formula

Have you ever wondered how to find the balancing point of a triangle? The answer can be found in this lesson on the centroid of a triangle! We'll learn how medians are used to locate the centroid and then discuss its various properties.

### 14. Flow Proof in Geometry: Definition & Examples

When someone claims something is true, it's important that they are able to support the claim with proof. In this lesson, you will learn how flow proofs are used in geometry to support a mathematical claim.

### 15. Infinite Sequence: Definition & Examples

After completing this lesson, you will be able to identify an infinite sequence and different types of infinite sequences. You will also be able to write additional terms of an infinite sequence and write general rules for infinite sequences.

### 16. Law of Detachment in Geometry: Definition & Examples

This lesson is about a fundamental logical argument type called the Law of Detachment. The definition is given, and examples of valid and invalid uses of it are provided.

### 17. Phase Shift: Definition & Formula

In mathematics, shifting a function horizontally is referred to as a phase shift. It is one of many ways that we can transform a function. In this lesson, we define phase shift and give a formula for determining its value.

### 18. Side-Side-Side (SSS) Triangle: Formula & Theorem

Side-side-side triangles are often found in geometric proofs. In this lesson, learn about the side-side-side postulate, and review what you've learned with a quiz.

### 19. Square Pyramid: Definition & Properties

This lesson provides an informative overview of the properties of square pyramids. Along with providing the definition for a square pyramid, the lesson includes the formulas for finding base area, volume, and surface area.

### 20. What is an Acute Triangle? - Definition, Facts & Example

In this lesson, you'll find out what makes a triangle an acute triangle. You'll also find some examples of what keeps a triangle from being acute. You'll also get to test your knowledge with a brief quiz.

### 21. Converse of a Theorem: Definition & Examples

After reading this lesson, you will know how to find the converse of mathematical theorems. We will also discover how the converse of a theorem is not always true.

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### Other Chapters

Other chapters within the High School Geometry: Tutoring Solution course

- Foundations of Geometry: Tutoring Solution
- Logic in Mathematics: Tutoring Solution
- Introduction to Geometric Figures: Tutoring Solution
- Properties of Triangles: Tutoring Solution
- Parallel Lines and Polygons: Tutoring Solution
- Similar Polygons: Tutoring Solution
- Quadrilaterals: Tutoring Solution
- Symmetry, Similarity and Congruence
- Circular Arcs and Circles: Tutoring Solution
- Conic Sections: Tutoring Solution
- Geometric Solids: Tutoring Solution
- Analytical Geometry: Tutoring Solution
- Introduction to Trigonometry: Tutoring Solution