# Ch 43: NES Middle Grades Math: Triangle Theorems & Proofs

### About This Chapter

## NES Middle Grades Math: Triangle Theorems & Proofs - Chapter Summary

Triangle proofs deserves its own chapter, and here you have it. Here we help you prepare for the National Evaluation Series (NES) Middle Grades Mathematics exam by giving in-depth instruction on the critical theorems you are likely to be tested on. The proofs and theorems described here include:

- Congruence postulates and proofs
- Similarity transformations
- AAS, HA, and HL theorems
- Perpendicular bisector and angle bisector theorems
- Congruency in right and isosceles triangles

Our talented math instructors walk you through each theorem and/or proof, providing useful real-world examples to help drive the principles home. The video lessons contain fun, original graphics to give you plentiful mental queues to draw upon on test day. Be sure to take the quizzes and chapter test for practice answering the kinds of questions you are likely to experience on the real exam.

### NES Middle Grades Math: Triangle Theorems & Proofs Chapter Objectives

This chapter is one of several that focuses on the Measurement and Geometry content domain of the NES Middle Grades Mathematics exam. This domain accounts for 25% of the test content and includes Objective 0008, which focuses on Euclidean geometry. There are 150 multiple-choice questions overall in the exam, which you must answer in less than 4 hours and 15 minutes. The passing score is the national benchmark score of 220 on a scale of 100-300. Though you will have to wait a couple of weeks to get your score report, you will know immediately after the test whether you passed. Check out the NES website for more detailed test information, testing dates/locations, and to receive your detailed score report.

### 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. Similarity Transformations in Corresponding Figures

Watch this video lesson to learn how you can tell if two figures are similar by using similarity transformations. Learn how to find the corresponding sides and angles and then how to compare them.

### 6. How to Prove Relationships in Figures using Congruence & Similarity

In this lesson, we'll look at similar and congruent figures and the properties that they hold. We will then look at how to use these properties to prove relationships in these figures in various examples.

### 7. Practice Proving Relationships using Congruence & Similarity

In geometry, if two shapes are similar they have the same shape but different sizes, while two congruent shapes have the same shape and size. In this lesson, you will learn how to prove that shapes are similar or congruent.

### 8. 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.

### 9. 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.

### 10. 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.

### 11. 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.

### 12. 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.

### 13. 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.

### 14. 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.

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### Other Chapters

Other chapters within the NES Middle Grades Mathematics (203): Practice & Study Guide course

- NES Middle Grades Math: Basic Arithmetic Operations
- NES Middle Grades Math: Solving Word Problems
- NES Middle Grades Math: Properties of Numbers
- NES Middle Grades Math: Estimating & Rounding
- NES Middle Grades Math: Number Theory
- NES Middle Grades Math: Roots & Radical Expressions
- NES Middle Grades Math: Exponents & Exponential Expressions
- NES Middle Grades Math: Scientific Notation
- NES Middle Grades Math: Simplifying Expressions
- NES Middle Grades Math: Integers
- NES Middle Grades Math: Rational Numbers
- NES Middle Grades Math: Introduction to Fractions
- NES Middle Grades Math: Operations with Fractions
- NES Middle Grades Math: Introduction to Decimals
- NES Middle Grades Math: Operations with Decimals
- NES Middle Grades Math: Percents
- NES Middle Grades Math: Money & Consumer Math
- NES Middle Grades Math: Number Patterns & Sequences
- NES Middle Grades Math: Properties of Functions
- NES Middle Grades Math: Graphing Functions
- NES Middle Grades Math: Negative Numbers
- NES Middle Grades Math: Absolute Value
- NES Middle Grades Math: Inequalities
- NES Middle Grades Math: Factoring
- NES Middle Grades Math: Monomials & Polynomials
- NES Middle Grades Math: Writing Algebraic Expressions
- NES Middle Grades Math: Basic Algebraic Expressions
- NES Middle Grades Math: Algebraic Distribution
- NES Middle Grades Math: One-Step Equations
- NES Middle Grades Math: Two-Step Equations
- NES Middle Grades Math: Rational Expressions
- NES Middle Grades Math: The Coordinate Graph
- NES Middle Grades Math: Linear Equations
- NES Middle Grades Math: Systems of Linear Equations
- NES Middle Grades Math: Rates, Ratios & Proportions
- NES Middle Grades Math: Measurement
- NES Middle Grades Math: Perimeter & Area
- NES Middle Grades Math: Polyhedrons & Geometric Solids
- NES Middle Grades Math: Symmetry, Similarity & Congruence
- NES Middle Grades Math: Properties of Lines
- NES Middle Grades Math: Angles
- NES Middle Grades Math: Triangles
- NES Middle Grades Math: Similar Polygons
- NES Middle Grades Math: The Pythagorean Theorem
- NES Middle Grades Math: Quadrilaterals
- NES Middle Grades Math: Circles
- NES Middle Grades Math: Circular Arcs & Measurement
- NES Middle Grades Math: Transformations
- NES Middle Grades Math: Data & Graphs
- NES Middle Grades Math: Statistics
- NES Middle Grades Math: Data Collection
- NES Middle Grades Math: Samples & Populations
- NES Middle Grades Math: Probability
- NES Middle Grades Math: Sets
- NES Middle Grades Mathematics Flashcards