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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…]

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 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…]

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…]

Applying the Transversal Theorems

When you cross two lines with a third line, the third line is called a transversal. You can use the transversal theorems to prove that angles are congruent or supplementary. [more…]

Using Auxiliary Lines in Proofs

The following proof introduces you to a new idea: adding a line or segment (called an auxiliary line) to a proof diagram to help you do the proof. Some proofs are impossible to solve until you add a line [more…]

Properties of Parallelograms

The properties of the parallelogram are simply those things that are true about it. These properties concern its sides, angles, and diagonals.

The parallelogram has the following properties: [more…]

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