5. Insert the proper preposition:

Euclidean geometry is an axiomatic system, … which all theorems ("true statements") are derived … a number of axioms. We are not only told that certain things exist, but are also given methods … creating them. In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects … saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.

… the beginning of the first book of the Elements, Euclid gives five postulates (axioms): Any two points can be joined … a straight line. Any straight line segment can be extended indefinitely … a straight line. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre. All right angles are congruent. Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other … that side if extended far enough. For example, if a triangle is constructed … … three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. For example a Euclidean straight line has no width, but any real drawn line will. To the ancients, the parallel postulate seemed less obvious than the others; verifying it physically would require us to inspect two lines to check that they never intersected, even … some very distant point, and this inspection could potentially take an infinite amount … time.

6. Translate into Ukrainian, and analyze the forms of the Subjunctive Mood

If this book were interesting, it would be in great demand.

If he’d come ten minutes later, he’d missed the train.

If you had worked harder last year, you wouldn’t have made these mistakes.

If she had more experience, she’d do better.

If I’d known that, I’d have put off the appointment.

6. If equals are added to equals, then the wholes are equal.

7. If equals are subtracted from equals, then the remainders are equal.

8. If the parallel postulate is dropped from the list of axioms without replacement, the result is the more general geometry called absolute geometry.

9. If a parabola is moving with exact velocity, it will never return because the curve reaches to infinity.

10. If the bodies fall to the earth with constant acceleration, they will move in a parabolic path.

7. Complete the following questions and answer them:

What would you do if to be in trouble..?

to be deceived..?

to be late to the meeting..?

2. How would you behave if to be responsible..?

not to ask for advice..?

not to find the necessary information..?

3. What would have happened if not to make proper arrangements..?

not to keep in a cold place..?

can’t attend to..?

4. What would you have done if to mention..?

to fail the exam..?

can’t keep the promise..?