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…]
How to Solve a Similar Polygon Problem
Recall that similar polygons are polygons whose corresponding angles are congruent and whose corresponding sides are proportional. The figure below shows similar pentagons, [more…]
How to Prove Triangles Similar Using the AA Theorem
You can use the AA (Angle-Angle) method to prove that triangles are similar. The AA theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles [more…]
How to Prove Triangles Similar with SSS~
You can prove that triangles are similar using the SSS~ (Side-Side-Side) method. SSS~ states that if the ratios of the three pairs of corresponding sides of two triangles are equal, then the triangles [more…]
How to Prove Triangles Similar with SAS~
You can prove that triangles are similar using the SAS~ (Side-Angle-Side) method. SAS~ states that if two sides of one triangle are proportional to two sides of another triangle and the included angles [more…]
How to Use CASTC after Proving Triangles Similar
CASTC is simply an acronym that stands for ‘Corresponding angles of similar triangles are congruent.’ You often use CASTC in a proof immediately after proving triangles similar [more…]
How to Use the Angle-Bisector Theorem
The Angle-Bisector theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the other two sides. The following figure illustrates [more…]
How to Identify Radii, Chords, and Diameters
When you work with circles, there are three straight-line components that you need to be able to identify: radii, chords, and diameters. [more…]
What You Need to Know About a Circle's Radius and Chords
When you’re working with a circle, there are five important theorems that you need to know about the properties of the circle’s radii and chords. (There are really just three ideas; but two of the theorems [more…]
Six Important Circle Theorems
The six circle theorems discussed here are all just variations on one basic idea about the interconnectedness of arcs, central angles, and chords (all six are illustrated in the following figure): [more…]
How a Tangent Relates to a Circle
A line is tangent to a circle if it touches it at one and only one point. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Check out the bicycle [more…]
How to Determine the Length of an Arc
An arc’s length means the same commonsense thing length always means — you know, like the length of a piece of string (with an arc, of course, it’d be a curved piece of string). Make sure you don’t mix [more…]
How to Solve Problems with the Altitude-0n-Hypotenuse Theorem
In a right triangle, the altitude that’s perpendicular to the hypotenuse has a special property: it creates two smaller right triangles that are both similar to the original right triangle. [more…]
How to Use Extra Radii to Solve a Problem
When buying a home, the three most important things to consider are location, location, location. With circles, it’s radii, radii, radii. In circle problems, you often need to add extra radii and partial [more…]
How to Determine the Measure of an Angle whose Vertex Is Inside a Circle
An angle that intersects a circle can have its vertex inside, on, or outside the circle. This article covers angles that have their vertex inside a circle—so-called [more…]
Geometry Symbols You Should Know
Using geometry symbols will save time and space when writing proofs, properties, and figuring formulas. The most commonly used geometry symbols and their meanings are shown below. [more…]
Proof Strategies Summarized
The following strategies can help you a great deal when you’re working on two-column geometry proofs. You should review these strategies and practice using them until they become internalized. These strategies [more…]
