Question

The solution must be complete, with explanations that are based on already studied facts, formulas, definitions, axioms, theorems and consequences from them.

All assignments require drawing.



Exercise 1.

Rectangle ABCD is given. The line a is parallel to AD and does not lie in the plane of the rectangle.

a) Prove that a||BC (7 points).

b) Prove that lines a and BD are intersecting (7 points).

c) Determine the cosine of the angle between the lines a and BD, if AB = 18 cm, BC = 24 cm. Justify your answer (15 points).



Task 2.

Given a rectangular box MNKLM1N1K1L1. The point E lies on the edge KK1, the point G lies on the edge NK, and the point F lies on the bottom face.

a) Construct a section of a parallelepiped by three given points E, F, G. Explain the construction of each of the segments (22 points).

b) Indicate the name (type) of the resulting polygon and shade its inner part (8 points).

1



Task 3.

An isosceles triangle ABC with base AB is given. From a point S lying outside the plane ABC, a perpendicular is dropped to point C.

a) Draw the linear angle of the dihedral angle SABC. Explain why this particular angle is the linear angle of the dihedral angle (16 points).

b) Find SC if AC = BC = 15 cm, AB = 18 cm, dihedral angle SABC is 30° (25 points).

Answer 1

and the point F lies on the bottom face.

a) Construct a section of a parallelepiped by three given points E, F, G. Explain the construction of each of the segments (22 points).

b) Indicate the name (type) of the resulting polygon and shade its inner part (8 points).

1

Task 3.

An isosceles triangle ABC with base AB is given. From a point S lying outside the plane ABC, a perpendicular is dropped to point C.

a) Draw the linear angle of the dihedral angle SABC. Explain why this particular