Regular Polygons. Special Quadrilaterals

In this chapter we’ll be mostly concerned with studying plane figures called polygons. Polygons are many-sided figures, with sides that are line segments. These simple closed figures are named according to the number of sides and angles they have, and may be classified by the measures of the angles or the measures of the sides. The simplest polygon is a triangle, a geometric plane figure having three sides. We have no difficulty listing all the polygons having up to ten sides. In the picture below you can see some of them.

A polygon is called regular if all of its sides and all of its interior angles are congruent. For instance, a square is a regular quadrilateral having four right angles and four equal sides.

It should be pointed out that the process of constructing a regular polygon is closely related to division of a circle into congruent parts. Students of mathematics will remember two major theorems concerning this problem.

Theorem. If a circle is divided into a certain number (greater than 2) of congruent parts, then:

(1) connecting every two consecutive division points by chords, we obtain a regular polygon inscribed into the circle.

(2) Drawing tangents to the circle at all the division points and extending each of them up to the intersection points with the tangents at the nearest division points we obtain a regular polygon circumscribed about the circle.

To find the sum of the interior angles of any polygon one can use the formula sum of the angles = (n – 2) 180º (where n stands for the number of sides or angles). For a triangle the sum is 180º. By drawing all diagonals from one single vertex of a polygon we can separate it into triangles. If you look back at the formula, you will see that n – 2 gives the number of triangles in the polygon, and that number is multiplied by 180º, which is the sum of the measures of all the interior angles in a triangle.

To find the perimeter of a regular polygon you should multiply the length of the sides by the number of sides.

Now let us dwell on polygons called quadrilaterals. In Euclidean plane geometry a quadrilateral is a polygon with four sides and four vertices or corners. Quadrilaterals are simple (not self-intersecting) or complex

(self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave. A kite is a convex trapezium that has two congruent pairs of adjacent sides. A deltoid is a concave trapezium.

The five most common types of quadrilaterals are the parallelogram, the rectangle, the square, the trapezoid and the rhombus. All quadrilaterals have some things in common. All of them 1) have four sides, 2) are coplanar, 3) have two diagonals and 4) the sum of their four interior angles equals 360º.

In addition, some of quadrilaterals possess special properties. As an example let’s take a parallelogram. Its special properties are as follows:

· Opposite sides are parallel. · Consecutive pairs of angles are supplementary.

· Opposite angles are congruent. · Opposite sides are congruent. · Diagonals bisect each other.

Below is a summary of the types of quadrilaterals. You might think of a quadrilateral like this: every square is a rectangle, but not every rectangle is a square. A rectangle is also a parallelogram, but a parallelogram may not be a rectangle.

Such a classification may prove helpful. It is worth remembering.

Ex. 7. Answer the following questions.

1. What is a polygon?

2. In what way do we classify polygons?

3. What polygon is called regular?

4. How is the process of constructing a regular polygon related to division of a circle into congruent parts?

5. Is it possible to obtain a regular polygon inscribed into the circle? (a regular polygon circumscribed about the circle)

6. Which formula is used for finding the sum of the interior angles of any polygon?

7. How can we find the perimeter of a regular polygon?

8. What are the most common types of quadrilaterals?

9. How many things do all quadrilaterals have in common?

10. What special properties does a parallelogram possess?

11. Do you know any other classifications of quadrilaterals? Are they worth remembering?

Ex. 8. Match the English words and word combinations with the Russian equivalents.

1. to draw diagonals 2. from one single vertex 3. in addition to 4. to circumscribe about the circle 5. adjacent sides 6. to draw tangents to the circle 7. a convex or a concave trapezium 8. stands for 9. special quadrilaterals 10. regular polygons 11. supplementary angles 12. bisect each other 13. the measure of the sides 14. a plane figure 15. possess properties 16. a consecutive pair 17. to inscribe into the circle 18. complex or crossed quadrilaterals

a. делят друг друга пополам b. специальные четырехугольники c. дополнительные углы d. плоская фигура e. вписать в окружность f. описать вокруг окружности g. означает h. выпуклая или вогнутая трапеция i. последовательная пара j. начертить диагонали k. провести касательные к окружности l. обладают свойствами m. кроме того n. из одной вершины o. смежные стороны p. сложные или пересекающиеся четырехугольники q. величины сторон r. правильные многоугольники

Ex. 9. Fill in the blanks with the words from the box. Mind there are two extra words.

a. rectangle b. regular c. vertices d. line segments e. corners f. vertex g. convex

h. concave i. special j. product k. rhombus l. dimensions m. trapezoid n. congruent

o. interior length p. quadrilateral q. diagonals r. number s. inscribed t. circumscribed u. kite

1. A simple closed figure formed by... is called a polygon.

2. In Euclidean plane geometry, a... is a polygon with four sides and four... or....

3. The area of a rectangle figure is the... of its....

4. A shape that is both a... and a... is a square (four equal sides and for equal angles).

5. A polygon is called... if all of its sides and all of its interior angles are....

6. Applying these geometric theorems we can obtain both a regular polygon... into the circle and a regular polygon... about the circle.

7. Finding the sum of the... angles of a polygon is not difficult.

8. If you wish to find the perimeter of a regular polygon you should multiply... of the sides by the... of the sides.

9. By drawing all … from one single … of a polygon we can separate it into triangles.

10. Simple quadrilaterals are either … or ….

11. A parallelogram possesses … properties.

Ex. 10. Guess what figure possesses the following properties and memorize them (a square, a trapezoid, a kite, a rectangle, a parallelogram, a rhombus).

1. A... has two parallel pairs of opposite sides.

2. A... has two pairs of opposite sides parallel, and four right angles. It is also a parallelogram, since it has two pairs of parallel sides.

3. A... has two pairs of parallel sides, four right angles, and all four sides are equal. It is also a rectangle and a parallelogram.

4. A... is defined as a parallelogram with four equal sides. It does not have to have 4 right angles.

5.... only has one pair of parallel sides. It's a type of quadrilateral that is not a parallelogram.

6.... has two pairs of adjacent sides that are equal.

Ex. 11. Find out whether the statements are True or False. Use the introductory phrases:

I think it is right. Quite so. Absolutely correct. I quite agree to it.

I am afraid it is wrong. I don’t quite agree to it. On the contrary. Far from it.

1. Plane figures bounded by four sides are called triangles.

2. The area of a square is the product of the length of its sides.

3. A rectangle is a parallelogram that has four obtuse angles.

4. Every interior angle in a convex polygon has a measure greater than 180⁰.

5. Rhombus is a parallelogram with four congruent angles.

6. Every square is a rectangle, but not every rectangle is a square.

7. To find the perimeter of a regular polygon, divide the length of the sides by the number of sides.

8. The area of a geometric figure is a quantity expressed by negative numbers.

9. A quadrilateral is a square if and only if it is both a rhombus and a rectangle.

10. The base and the altitude of a rectangle are called its dimensions.

Ex. 12. Ask special questions using the question words in brackets.

Venn Diagram

1. Let us use a Venn diagram to group the types of quadrilaterals (why).

2. A Venn diagram uses overlapping (частично совпадающие) circles. It shows relationships between groups of objects (what). 3. All quadrilaterals can be separated into three sub-groups: general quadrilaterals, parallelograms and trapezoids (how many).

4. Since all four sides of a rectangle don't have to be equal a rectangle isn’t always a rhombus (why). 5. However, the sets of rectangles and rhombuses intersect (which). 6. Their intersection is the set of squares (whose). All squares are both a rectangle and a rhombus.

7. We can put squares in the intersection of the two circles (where). 8. From this diagram, you can see that a square is a quadrilateral, a parallelogram, a rectangle, and a rhombus (from what).

9. A trapezoid isn’t a parallelogram because it has only one pair of parallel sides (how many). 10. That is why we must show the set of trapezoids in a separate circle on the Venn diagram (in what way).

Let’s consider kites. 11. Kites are quadrilaterals that can be parallelograms. (what type of). 12. If its two pairs of sides are equal, it becomes a rhombus

(in what case).

Ex. 13. Choose the best alternative to the English sentence.

1. They’re used to coming back late.

a. Они часто возвращались домой поздно.

b. Они привыкли возвращаться поздно.

c. Они воспользовались своим поздним возвращением.

2. I can’t help doing it myself.

a. Ничем не могу помочь, так как делаю это сама.

b. Не может быть, что я сделала это сама.

c. Не могу не сделать это сама.