Учебное пособие. / Сост

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Тип	Учебное пособие

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Read the sentences and think of a word which best fits each space.

a) 1. The Egyptians were mostly concerned with applying … to their everyday problems.

2. In 300 B.C. all the known facts about Greek geometry were put into a logical sequence by … .

3. Today geometry includes not only the study of the … and … of the earth and all things on it, but also the study of relations between geometric … .

4. The most fundamental idea in the study of geometry is the idea of a … .

You cannot see a … , feel a … , or move a … , because it has no dimensions.

For ideas: shape, point (4), size, geometry, Euclid, object.
b) 1. ... are generally studied as graphs of equations using a coordinate systems.

2. Only ... curves (or arcs) have end points.

3. A curve that does entirely in a plane is a ... curve.

4. A curve that does not lie in ... is a skew or twisted curve.

5. The rate of change of direction of a curve at a particular point on that curve is called a ...

6. The angle δψ through which the tangent to a curve moves as the point of contact moves along any arc is the ... of this arc.

7. The ... of any arc is defined as the total curvature divided by the arc length.

8. The circle of curvature at any point on a curve is the circle that is ... to the curve at that point.

9. There are two ... in which the radius of curvature has an absolute maximum and absolute minimum.

10. The principal curvatures at the point are the curvatures in two ... directions.
TEXT II.
ANGLES
An angle is a configuration of two lines (the sides or arms) meeting at a point (the vertex). Often an angle is regarded as the measure of rotation involved in moving from one initial axis to coincide with another final axis (termed a directions angle). If the amount and sense of the rotation are specified the angle is a rotation angle, and is positive if measured in an anticlockwise sense and negative if in a clockwise sense.

Angles are classified according to their measure:

Null (or zero) angle - zero rotation (0°).
Right angle - a quarter of a complete turn (90°)
Flat (or straight) angle - half a complete turn (180°).
Round angle (or perigon) - one complete turn (360°),
Acute angle - between 0° and 90°.
Obtuse angle - between 90° and 180°.
Reflex angle - between 180°and 360°.
The angle of elevation of a point A from another point В is the angle between the line AB and the horizontal plane through B, with A lying above the plane. The angle of depression is similarly defined with A lying below the plane. The angle at point В made by lines AB and CB is denoted by ABC.

Read and decide which of the statements are true and which are false.

Change the sentences so they are true.
1. An angle is often regarded as the measure of rotation involved in moving from one initial axis to coincide with another final axis.

2. There’re eleven types of angles in their classification according to their measure.

3. 90° - it is the measure of an acute angle.

4. An angle is positive if it is measured in a clockwise sense.

5. The measure of a reflex angle is between 180°and360°.

6. The main difference of an angle of elevation of a points and its angle of depression is the following one: in the case of the angle of elevation the point A lies above the plane and in the case of the angle of depression - below the plane.

Match the terms from the left column and definitions

from the right column:

an angle	formed by, or with reference to, a straight line or plane perpendicular to a base
null	of less than 90 degrees
right	designating an angle greater than a straight angle (180 degrees)
obtuse	height above a surface, as of the earth
flat	the shape made by two straight lines meeting at a common point, the vertex, or by two planes, meeting along an edge
acute	a decrease in force, activity, amount, etc. - a decrease in force, activity, amount, etc.
reflex	greater than 90 degrees and less than 180 degrees greater than 90 degrees and less than 180 degrees
elevation	designating of, or being zero, as: a) having all zero elements (null matrix), b) having a limit of zero (null sequence), c) having no members whatsoever (null set)
depression	absolute, positive

III. Give the Russian equivalents of the following words and word combinations:

side (arm)
acute angle
angle of depression
direction angle
sense of rotation
clockwise sense
vertex
obtuse angle
rotation
reflex angle
rotation angle
angle of elevation
right angle
flat (straight) angle
round angle (perigon)
null (zero) angle

Give the English equivalents of the following words

and word combinations:
тупой угол, развёрнутый угол, нулевой угол, угол возвышения, угол понижения, прямой угол, полный угол, сторона, направление вращения, вершина, угол в пределах от 180° 360, вращение (поворот), острый угол, по часовой стрелке, против часовой стрелки, угол вращения, направляющий угол.

V. Translate the sentences into English.
1. Если две стороны и угол между ними одного треугольника равны соответственно двум сторонам и углу между ними другого треугольника, то такие треугольники равны.

2. Две прямые называются перпендикулярными, если они пересекаются под прямым углом.

3. Какой угол называется прилежащим?

4. Докажите, что вертикальные углы равны.

5. Сумма трех этих углов равна 270°.
VI. Read the sentences and think of a word which best fits each space.
1. An angle is a ... of two lines (the sides or …) meeting at a point called the vertex.

2. Flat (or ...) angle means half a ... turn.

3. An obtuse angle is greater than an ... angle.

4. The measure of a ... angle is between 180°and 360°.

5. Angles are classified according to their....

6. Clockwise means the ... in which the hands of a clock rotate.

7. The largest angle is the ... angle being 360 degrees.

Answer the questions on the text "Angles".

What is an angle?
Can one say that an angle is regarded as the measure of rotation involved in moving from one initial axis to coincide with another final axis?
What are characteristics of a null angle?
An acute angle is an angle between 0° and 90°, isn't it?
What are characteristics of an obtuse angle?
What are characteristics of a reflex angle?
Is there any difference between the angle of depression and the angle
of elevation?

TEXT III.
A POLYGON
A polygon is a figure formed by three or more points (vertices) joined by line segments (sides). The term is usually used to denote a closed plane figure in which no two sides intersect. In this case the number of sides is equal to the number of interior angles. If all the interior angles are less than or equal to 180°, the figure is a convex polygon; if it has one or more interior angles greater than 180°, it is a concave polygon, A polygon that has all its sides equal is an equilateral polygon; one with all its interior angles equal is an equiangular polygon. Note that an equilateral polygon need not be equiangular, or vice versa, except in case of an equilateral triangle. A polygon that is both equilateral and equiangular is said to be regular. The exterior angles of a regular polygon are each equal to 360° /n, where n is a number of sides.

The distance from the center of a regular polygon to one of its vertices is called the long radius, which is also a radius of the circumcircle of the polygon. The perpendicular distance from the center to one of the sides is called the short radius or apothem, which is also the radius of the inscribed circle of the polygon.

A regular star polygon is a figure formed by joining every m-th point, starting with a given point, of the n points that divide a circle's circumference into n equal parts, where m and n are relatively prime, and n is equal two or greater than 3. This star polygon is denoted by {m/n}. When m = 1, the resulting figure is a regular polygon. The star polygon {5/2} is the pentagram.

Read and decide which of the statements are true and which are false.

Change the sentences so they are true.
1. A polygon is a figure formed only by three vertices joined by line segments.

2. No sides of a polygon usually intersect.

3. All the interior angles of a convex polygon are greater than or equal to180.

4. An equilateral polygon is a polygon whose no sides are equal.

5. An equiangular polygon is a polygon whose all interior angles are equal.

6. The regular polygon is both equilateral and equiangular.

7. The perpendicular distance from the center to one of the sides is called the long radius.

8. The long radius is also the radius of the inscribed circle of the polygon.

9. The star polygon is usually denoted by {m/n}.

Match the terms from the left column and definitions from the right column:

apothem	to draw a figure inside another figure so that their boundaries touch at as many points as possible
circle	a) at right angles to a given plane or line, b) exactly upright; vertical; straight up or down
convex	a) any of the four angles formed on the inside of two straight lines when crossed by a transversal, b) the angle formed inside a polygon by two adjacent sides
concave	any figure of five lines
equilateral	a geometrical figure having three angles and three sides
equiangular	a) the point of intersection of the two sides of an angle, b) a comer point of a triangle, square, cube, parallelepiped, or other geometric figure bounded by lines, planes, or lines and planes
exterior angle	curving outward like the surface of a sphere
to inscribe	having all angles equal
interior angle	a closed plane figure, esp. one with more than four sides and angles
pentagram	a) any of the four angles formed on the outside of two straight lines when crossed by a transversal, b) an angle formed by any side of a polygon and the extension of the adjacent side
perpendicular	a plane figure bounded by a singly curved line, every point of which is equally distant from the point at the center of the figure
polygon	hollow and curved like the inside half of a hollow ball
triangle	having all sides equal
vertex	the perpendicular from the center of a regular polygon to any one of its sides

Give the Russian equivalents of the following words and word

combinations:

exterior angle
circumcircle
perpendicular
apothem
inscribed circle
star polygon
relatively prime
regular star polygon
pentagram
closed plane figure
interior angle
convex polygon
concave polygon
equilateral polygon
equiangular polygon
equilateral triangle
regular polygon
long radius

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