Geometry Quadrilaterals

A quadrilateral is a two-dimensional figure with four sides. The Latin word “latus” means side. (A bi-lateral agreement is an agreement between two countries, as opposed to multi-lateral, meaning between many countries.) Quattor means four in Latin. (It is the root for many English words, like quad, quadriceps and quadrangle.)

A quadrilateral is a closed figure, which means the lines join up. All four sides are straight. A quadrilateral is a polygon. A polygon is a closed figure with three or more straight sides. The word "polygon" derives from the Greek adjective πολύς (polús) "much", "many" and γωνία (gōnía) "corner" or "angle“; so you could define it as a closed, two-dimensional figure with many angles. Some people call a quadrilateral a tetragon, which would be in line with calling a polygon with 5 angles a pentagon.

A regular figure is a polygon with all sides equal and all angles equal. Another way of saying this is a polygon with congruent sides and angles. A square is a regular quadrilateral. All other quadrilaterals are irregular.

We give special names to a number of quadrilaterals. Those names depend on their attributes. An attribute is a characteristic to describe an object. In the context of geometry attribute usually refers to the shape, size, or line direction. In a different context it can also refer to colour. The attribute of an angle is usually it’s size: acute, right, obtuse, straight or reflexive. It can also be congruent to another angle, which means: the same size. The attribute of a line is usually whether it is congruent with another line or parallel to it. Line segments are congruent if they have the same length. They need not be parallel to be congruent. Lines that are next to one another are adjacent.

A diagonal is a line segment that goes from one vertex to another.

A quadrilateral has two diagonals that intersect. In some quadrilaterals each diagonal cuts the other diagonal in half. This is called mutually bisecting diagonals. In other quadrilaterals the diagonals intersect at a 90 degree angle. This is called perpendicular diagonals.

The sum of the interior angles of a quadrilateral is always 360 ⁰.

A square is a regular quadrilateral. It has four congruent angles, which by definition must be 360 : 4 = 90⁰ It has two sets of parallel sides. It has four congruent sides.

If we lengthen two opposite sides of the square by the same amount, we create a rectangle. A rectangle is an irregular quadrilateral where every angle is a right angle and the opposite sides are parallel. It has four congruent right angles. It has two sets of congruent and parallel sides.

A square is a regular quadrilateral. It has four congruent right angles. It has two sets of parallel sides. It has four congruent sides.

When the angles are not right angles, but the sides remain the same, the quadrilateral becomes a rhombus. A rhombus is a four-sided shape where all sides have equal length. It is irregular. Also opposite sides are parallel and opposite angles are congruent. Another interesting thing is that the diagonals meet in the middle at a right angle. They have perpendicular diagonals. In popular language a rhombus is sometimes called a diamond, but you should not use that word in math.

If you take a rhombus and lengthen two of its sides, you get a parallelogram. If you take a rectangle and change one set of angles, you also get a parallelogram. A parallelogram has opposite sides parallel and equal in length. Also opposite angles are congruent. (angles "a" are the same, and angles "b" are the same). The diagonals of a parallelogram are mutually bisecting, but not perpendicular.

A square is a special kind of rhombus, rectangle and parallelogram. When you see a square and call it a parallelogram, you are not wrong, but it is better to call it a square, just like it is more specific to call a Blue Jay a Blue Jay and not a bird or an animal.

If you take a parallelogram and change the direction of one of the sides, you get a trapezoid. A trapezoid is a quadrilateral with one set of parallel sides. A trapezoid with one set of congruent angles is called an isosceles trapezoid. A trapezoid with two right angles is called a right trapezoid.

A kite is a quadrilateral with two pairs of congruent adjacent sides. Adjacent sides are sides that meet, or are next to one another. The adjacent sides are congruent, or equal in length. The angles are equal where the non congruent lines meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonals bisects (cuts equally in half) the other. When the figure looks like a kite in the sky, it is convex. It can also be concave and look like an arrow head.

Describe this quadrilateral using math language but without using the name of the shape.

We did this in class. Do you remember how to do it?

Why can’t this quadrilateral be called a square, a parallelogram, a rectangle, a rhombus, a trapezoid or a kite?

We did this in class. Do you remember how to do it?

Double check your answer by adding all the angles. It should add up to 360 ⁰.

These are questions 1 and 2 of page 91. To answer question 2 you need to know the definitions of the different quadrilaterals. Question 2a). You answer should start with: A trapezoid is a quadrilateral with two parallel sides.

This is question 4 of MMS 91. You need not do 4d.