Geometry

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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…]

Properties of Rhombuses, Rectangles, and Squares

The three special parallelograms — rhombus, rectangle, and square — are so-called because they’re special cases of the parallelogram. (In addition, the square is a special case or type of both the rectangle [more…]

The Properties of a Kite

A kite is a quadrilateral in which two disjoint pairs of consecutive sides are congruent (“disjoint pairs” means that one side can’t be used in both pairs). Check out the kite in the below figure. [more…]

How to Prove that a Quadrilateral Is a Parallelogram

There are five ways in which you can prove that a quadrilateral is a parallelogram. The first four are the converses of parallelogram properties (including the definition of a parallelogram). Make sure [more…]

How to Prove that a Quadrilateral Is a Rectangle

There are three ways to prove that a quadrilateral is a rectangle. Note that the second and third methods require that you first show (or be given) that the quadrilateral in question is a parallelogram [more…]

How to Prove that a Quadrilateral Is a Rhombus

You can use the following six methods to prove that a quadrilateral is a rhombus. The last three methods in this list require that you first show (or be given) that the quadrilateral in question is a parallelogram [more…]

How to Prove that a Quadrilateral Is a Square

There are four methods that you can use to prove that a quadrilateral is a square. In the last three of these methods, you first have to prove (or be given) that the quadrilateral is a rectangle, rhombus [more…]

How to Calculate the Area of a Parallelogram, Kite, or Trapezoid

The area formulas for the parallelogram, kite, and trapezoid are based on the area of a rectangle. The following figures show you how each of these three quadrilaterals relates to a rectangle, and the [more…]

A Rhombus Area Problem

Here’s a rhombus area problem involving triangles and ratios: Find the area of rhombus RHOM given that MBis 6 and that the ratio of RB to BH is 4 : 1, as shown in the following figure. [more…]

How to Calculate the Area of a Kite

You calculate the area of a kite by using the lengths of its diagonals. Here’s an example: what’s the area of kite KITE in the following figure? [more…]

How to Calculate the Area of a Trapezoid

You can use the right-triangle trick to find the area of a trapezoid. The following trapezoid TRAP looks like an isosceles trapezoid, doesn’t it? Don’t forget — looks can be deceiving. [more…]

The Properties of Trapezoids and Isosceles Trapezoids

A trapezoid is a quadrilateral with exactly one pair of parallel sides (the parallel sides are called bases). The following figure shows a trapezoid to the left, and an isosceles trapezoid on the right [more…]

How to Calculate the Area of a Regular Octagon

You can calculate the area of a regular octagon with the standard regular polygon method, but there’s a nifty alternative method based on the fact that a regular octagon is a square with its four corners [more…]

Interior and Exterior Angles of a Polygon

Everything you need to know about a polygon doesn’t necessarily fall within its sides. You may need to find exterior angles as well as interior angles when working with polygons: [more…]