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Geometry

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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…]

How to Find the Number of Diagonals in a Polygon

To find the number of diagonals in a polygon with nsides, use the following formula: [more…]

How to Identify and Name Similar Polygons

You can identify similar polygons by comparing their corresponding angles and sides. As you see in the following figure, quadrilateral WXYZ is the same shape as quadrilateral [more…]

How to Align Similar Polygons

If you get a problem with a diagram of similar polygons that aren’t lined up in the same orientation, consider redrawing one of them so that they’re both aligned in the same way. This may make the problem [more…]

Determining a Triangle's Area from Its Base and Height

If you know the base and height of a triangle, you can use a tried-and-true formula to find its area. You likely first ran into the basic triangle area formula in about sixth or seventh grade. If you’ve [more…]

Determining a Triangle's Area from Its Three Sides

When you know the length of a triangle’s three sides and you don’t know an altitude, you can use Hero’s formula to find the area. Check it out: [more…]

Determining the Area of an Equilateral Triangle

To calculate a triangle’s area — for most types of triangles — you need to know the triangle’s base and height. However, with an equilateral triangle, all you need to know is the length of one of its sides [more…]

How to Find the Centroid of a Triangle

The three medians of a triangle intersect at its centroid. The centroid is the triangle’s balance point, or center of gravity. (In other words, if you made the triangle out of cardboard, and put its centroid [more…]

How to Do a Parallelogram Proof

A good way to begin a proof is to think through a game plan that summarizes your basic argument or chain of logic. The following examples of parallelogram proofs show game plans followed by the resulting [more…]

How to Prove that a Quadrilateral Is a Kite

Proving that a quadrilateral is a kite is a piece of cake. Usually, all you have to do is use congruent triangles or isosceles triangles. Here are the two methods: [more…]

How to Calculate the Area of a Quadrilateral

There are five formulas that you can use to calculate the area of the seven special quadrilaterals. There are only five formulas because some of them do double duty — for example, you can calculate the [more…]

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