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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…]

Identifying Triangles by Their Angles

You can classify triangles by their angles as well as by their sides. The classifications based on angles are as follows: [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…]

Working with Pythagorean Triple Triangles

The first four Pythagorean triple triangles are the favorites of geometry problem-makers. These triples — especially the first and second in the list that follows — pop up all over the place in geometry [more…]

The Components of a Proof

A two-column geometry proof is a problem involving a geometric diagram of some sort. You’re told one or more things that are true about the diagram (the [more…]

The Transitive and Substitution Properties

You’re probably already familiar with the Transitive Property and the Substitution Property from algebra. If a = b and b = c, then a = c, right? That’s transitivity. And if [more…]

Completing the Square for Conic Sections

When the equation of a conic section isn't written in its standard form, completing the square is the only way to convert the equation to its standard form. The steps of the process are as follows: [more…]

How to Find the Incenter, Circumcenter, and Orthocenter of a Triangle

Every triangle has three “centers” — an incenter, a circumcenter, and an orthocenter — that are located at the intersection of rays, lines, and segments associated with the triangle: [more…]

Identifying the 45 – 45 – 90 Degree Triangle

A 45 – 45 – 90 degree triangle (or isosceles right triangle) is a triangle with angles of 45°, 45°, and 90° and sides in the ratio of [more…]

Identifying the 30 – 60 – 90 Degree Triangle

The 30 – 60 – 90 degree triangle is in the shape of half an equilateral triangle, cut straight down the middle along its altitude. It has angles of 30°, 60°, and 90° and sides in the ratio of [more…]

Using the Side-Side-Side Method to Prove Triangles Congruent

The SSS (Side-Side-Side) postulate states that if the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. The following figure illustrates [more…]

Using the Side-Angle-Side Method to Prove Triangles Congruent

The SAS (Side-Angle-Side) postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent [more…]

Using the Angle-Side-Angle Method to Prove Triangles Congruent

The ASA (Angle-Side-Angle) postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent [more…]

Using the Isosceles Triangle Theorems to Solve Proofs

The following two theorems — If sides, then angles and If angles, then sides — are based on a simple idea about isosceles triangles that happens to work in both directions: [more…]


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