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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…]

Bisecting and Trisecting Segments

Bisection and trisection involve cutting something into two or three equal parts. If you’re a fan of bicycles and tricycles and bifocals and trifocals — not to mention the biathlon and the triathlon, bifurcation [more…]

Bisecting and Trisecting Angles

The terms angle bisection and angle trisection describe two ways in which you can divide up an angle equally into two (or three) smaller, congruent angles. Their definitions are often used in proofs. [more…]

Geometry Formulas You Should Know

Below are several of the most important geometry formulas, theorems, properties, and so on that you use for solving various problems. If you get stumped while working on a problem and can’t come up with [more…]

How to Solve a Common-Tangent Problem

The common-tangent problem is named for the single tangent segment that’s tangent to two circles. Your goal is to find the length of the tangent. These problems are a bit involved, but they should cause [more…]

How to Solve Similar Triangle Problems with the Side-Splitter Theorem

You can solve certain similar triangle problems using the Side-Splitter Theorem. This theorem states that if a line is parallel to a side of a triangle and it intersects the other two sides, it divides [more…]

Using the Hypotenuse-Leg-Right Angle Method to Prove Triangles Congruent

The HLR (Hypotenuse-Leg-Right angle) theorem — often called the HL theorem — states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle [more…]

How to Do an Indirect Proof

Indirect proofs are sort of a weird uncle of regular proofs. With an indirect proof, instead of proving that something must be true, you prove it indirectly [more…]

Definitions and Theorems of Parallel Lines

Parallel lines are important when you study quadrilaterals because six of the seven types of quadrilaterals (all of them except the kite) contain parallel lines. The eight angles formed by parallel lines [more…]

Working with More than One Transversal

When a parallel-lines-with-transversal drawing contains more than three lines, identifying congruent and supplementary angles can be kind of challenging. The following figure shows you two parallel lines [more…]

Identifying the Seven Quadrilaterals

A quadrilateral is a polygon with four sides. There are seven quadrilaterals, some that are surely familiar to you, and some that may not be so familiar. Check out the following definitions and the quadrilateral [more…]

How Various Quadrilaterals Are Related

The quadrilateral family tree in the figure below shows you the relationships among the various quadrilaterals. The following list of questions gives you a taste of some of these relationships. [more…]


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