# Ch 5: High School Geometry: Triangles, Theorems and Proofs

### About This Chapter

## High School Geometry: Triangles Theorems and Proofs - Chapter Summary and Learning Objectives

Congruency merely means having the same measure. In this chapter, you can learn triangle congruence postulates and proofs, in addition to finding out how to prove relationships in figures using both similarity and congruence. Explore essential theorems related to triangles through several lessons in this chapter. Upon completion, you should feel comfortable working with:

- Similarity transformations
- Congruency of right triangles and isosceles triangles
- Congruence proofs
- Triangle congruence postulates

Video | Objective | |
---|---|---|

Applications of Similar Triangles | Learn how to solve applications of similar triangles. | |

Triangle Congruence Postulates | Describe the SAS, ASA and SSS postulates, providing examples of each. | |

Congruence Proofs | Define and use CPCTC to solve an applied problem. | |

Converse of a Statement | Identify the converse of a statement and explain how it must be proven to be true before it may be used as a reason in any proof. | |

Similarity Transformations | When given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. | |

Proving Relationships in Figures Using Congruence and Similarity | Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. | |

Practice Proving Relationships | Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. | |

The AAS Theorem | Prove the angle-angle-side theorem. | |

The HA Theorem | Prove the hypotenuse-angle theorem. | |

The HL Theorem | Prove the hypotenuse-leg theorem. | |

Segment Bisector Theorem | Prove the segment bisector theorem by showing an example giving the perpendicular bisector of a given line segment. | |

Angle Bisector Theorem | Prove the angle bisector theorem by showing an example giving the bisector of a given angle. | |

Congruency of Right Triangles | Define the LA and LL theorems. | |

Congruency of Isosceles Triangles | Show how the base angles of an isosceles triangle are congruent. |

### 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 Geometry: High School course

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