Geometry

What Is a Geometry Proof?

A geometry proof — like any mathematical proof — is an argument that begins with known facts, proceeds from there through a series of logical deductions, and ends with the thing you’re trying to prove. [more…]

Getting to Know the Five Simplest Geometric Objects

The study of geometry begins with the definitions of the five simplest geometric objects — point, line, segment, ray, and angle — as well as two extra definitions [more…]

Getting to Know Points

Although individual points have no features, when you group them, you can create several different types of points: collinear, non-collinear, coplanar, and non-coplanar. Each type merits an explanation [more…]

Getting to Know Lines

There are different types of lines (or segments or rays) or pairs of lines (or segments or rays). You can identify single lines based on the direction they’re pointing [more…]

Getting to Know Planes

When two geometric planes interact with each other, it is in one of two ways: as parallel planes or as intersecting planes. Here are the definitions for these two types of relationships between a pair [more…]

Getting to Know Angles

Angles are one of the basic building blocks of triangles and other polygons. There are five types of angles: acute, right, obtuse, straight, and reflex. You see angles on virtually every page of any geometry [more…]

Getting to Know Angle Pairs

Adjacent angles and vertical angles always share a common vertex, so they’re literally joined at the hip. Complementary and supplementary angles can share a vertex, but they don’t have to. Here are the [more…]

How to Measure Line Segments

To find the measure or size of a segment, you simply measure its length. What else could you measure? After all, length is the only feature a segment has. You’ve got your short, your medium, and your long [more…]

How to Measure Angles

Measuring angles is pretty simple: the size of an angle is based on how wide the angle is open. Here are some points and mental pictures that will help you to understand how angle measurement works. [more…]

Adding and Subtracting Segments and Angles

Adding and subtracting segments and angles isn’t exactly rocket science. But it is important because this geometric arithmetic comes up in proofs and other geometry problems. Here’s how it works: [more…]

Using If-Then Logic

Every geometry proof is a sequence of deductions that use if-then logic. You write one of the given facts as statement 1. Then, for statement 2, you put something that follows from statement 1 and write [more…]

Working with Definitions, Theorems, and Postulates

Definitions, theorems, and postulates are the building blocks of geometry proofs. With very few exceptions, every justification in the reason column is one of these three things. The below figure shows [more…]

How to Prove Angles Are Complementary or Supplementary

Complementary angles are two angles that add up to 90°, or a right angle; two supplementary angles add up to 180°, or a straight angle. These angles aren’t the most exciting things in geometry, but you [more…]

Using Addition Theorems in Proofs

There are four addition theorems: two for segments and two for angles. They are used frequently in proofs.

Use the following two addition theorems for proofs involving three segments or three angles: [more…]

Using Subtraction Theorems in Proofs

There are four subtraction theorems you can use in geometry proofs: two are for segments and two are for angles. Each of these corresponds to one of the addition theorems. [more…]

Using Theorems of Like Multiples and Like Divisions in Proofs

The Multiplication and division theorems are based on very simple ideas, but they do trip people up from time to time, so pay careful attention to how these theorems are used in the example proofs. [more…]

Proving Vertical Angles Are Congruent

When two lines intersect to make an X, angles on opposite sides of the X are called vertical angles. These angles are equal, and here’s the official theorem that tells you so. [more…]

Identifying Scalene, Isosceles, and Equilateral Triangles

Triangles are classified according to the length of their sides or the measure of their angles. These classifications come in threes, just like the sides and angles themselves. [more…]

Using The Triangle Inequality Principle

The triangle inequality principle states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This idea comes up in a fair number of problems, so [more…]

Finding the Altitude of a Triangle

The altitude of a triangle is a segment from a vertex of the triangle to the opposite side (or to the extension of the opposite side if necessary) that’s perpendicular to the opposite side; the opposite [more…]