Study program
First cycle study programme in mathematics (Bachelor
level)
1st cycle
Study level
Course title
Elementary geometry
Course code
MAT01-012
Language of instruction
English
Course description
Course objective. The objective of this course is to
systematise, consolidate and deepen the knowledge of the
elementary primary-school geometry, without giving
axiomatic of geometries. Classical geometrical contents
will be updated by demonstrations on computers.
Prerequisites. Not necessary.
Course contents.
1. Introduction to the planimetry. Basic objects of
geometry in plane (points and straight lines). Axioms of
Euclidean geometry plane. Axioms about paralleles. (The
axioms will be given only as information and dealt with
very elementary.)
2. Prominent sets of points in the plane. Half-line.
Segment. Convex sets in the plane. Half-plane. Angle.
Measure of angle. Vertical angles. Angles with parallel
arms and angles with perpendicular arms. Angles along
transversal. Triangle. Sum of angles in a triangle. Relation
of triangle. Quadrangle. Diagonal of a quadrangle.
Trapezoid. Parallelogram. Rhomb. Rectangle. Square.
Quadrangles with perpendicular diagonals. Multiangles.
Circumference and circle. (Only proofs referring to angles
will be dealt with in detail; all other concepts will be only
defined.)
3. Congruence of a triangle. Definition of triangle
congruence. Triangle congruence theorems. Perpendicular
bisector theorem. Four basic constructions of a triangle.
Characterisation of a parallelogram and a rhomb. The
midline of a triangle theorem. Four characteristic points of
a triangle. Circumcircle and incircle of a triangle. The
midline of a trapezoid theorem. Theorem about the
bisector of an angle.
4. Perimeter and area. Perimeter and area of a polygon.
Areas of square, parallelogram, triangle, trapezoid,
quadrangle with perpendicular diagonals. Heron's formula.
Connection between the area of a triangle and its sides and
the radius of its escribed circles. Area of a circle. Length
of a circumference.
5. Similarity of triangles. Thales' theorem of proportion.
Theorem about bisector of an interior angle in a triangle.
Definition of similarity of triangles. Pythagorean theorem
(some proofs) and its converse. Euclidean theorem.
6. Theorems about circumference. Theorem about
peripherical and central angle. Thales' theorem about
angle on a diameter. Circumscribed and inscribed
quadrilateral.
7. Plane mapping. Isometries of a plane. Axial and central
symmetry. Rotation. Translation. Homothety. Eulerean
line. Mapping of similarity.
8. Introduction to stereometry. Basic objects of geometry
of space (points, lines and planes). Axioms of Euclidean
geometry of space. Determination of plane and a line in
the space. Halfspace. Parallel lines and planes.
Perpendicular lines and planes. Theorem of three normals.
9. Angles between lines and planes. Angle between two
lines. Angle between line and plane. Angle between two
planes.
10. Distance in the space. Distance from point to plane.
Distance from point to line. The shortest distance between
skew lines. Symmetral planes of a segment and of a couple
of planes. Dihedrons and trihedrons.
11. Polyhedra. Idea of polyhedron. Some kinds of
polyhedra (pyramid, bipyramid, prism). Eulerean formula
for polyhedra. Regular polyhedra (Platonean bodies).
Volume and surface area of a polyhedron - rectangular
parallelepiped, parallelepiped, prism, pyramid and
truncated pyramid. Cavalieri's principle.
12. Round bodies. Cylindar. Cone. Sphere. Volume
and surface of round bodies - volume and surface area
of cylinder, cone, sphere.
Form of teaching
consultative teaching
Form of assessment
Students’ knowledge is assessed during the semester
through tests and homework. Written part of the final
examination can be replaced by tests.
Number of ECTS
5
Class hours per week
2+2+0
Minimum number of
students
Period of realization
summer semester
Lecturer
Zdenka Kolar-Begović, Associate Professor