Download MATH 8 ASSIGNMENT 4: EUCLIDEAN GEOMETRY Undefined

MATH 8
ASSIGNMENT 4: EUCLIDEAN GEOMETRY
FEB 3, 2013
Undefined notions
The key notions are: point, line, distance between points, measure (size) of an angle (see below)
We will also use the notions “lie between” and “on the same side”.
Using these (and Axioms below, we can define – ray AB: all the points on line AB which are on the
same side of A as B.
– segment AB: all the points on line AB which are between A, B
– Angle: a figure consisting of two rays with a common vertex. As usual, we define right angle, obtuse
and acute angles.
– midpoint of a segment AB: a point M on AB such that AM = M B
– angle bisector of angle ∠AOB: a ray OM such that ∠AOM = ∠M OB
Axioms
Axiom 1. For any two distinct points A, B, there is a unique line through these points.
Axiom 2. Distances add up: if B is on the line AC, between A and C, then AC = AB + BC
Axiom 3. Angles add up: if ray OB is inside the angle AOC, then ∠AOC = ∠AOB + ∠BOC
We will also use without proof various “obvious” properties of the notions “lie between” /“be inside”.
[In more rigorous courses, some of them are taken as axioms and the rest are proved, but we will be more
relaxed.]
Triangles
Triangle: a figure consisting of 3 distinct points (vertices) together with the segments connecting them.
Two triangles are called congruent 4ABC ∼
= 4A0 B 0 C 0 if the corresponding angles are equal and the
0
corresponding sides are equal: ∠A = ∠A , ∠A = ∠A0 , ∠B = ∠B 0 , ∠C = ∠C 0 , AB = A0 B 0 , AC = A0 C 0 ,
BC = B 0 C 0 . Note that when we write 4ABC ∼
= 4A0 B 0 C 0 , the order in which the vertices are written is
important!
Axiom 4. SAS congruence test for triangles: if ∠A = ∠A0 , AB = A0 B 0 , AC = A0 C 0 , then 4ABC ∼
=
4A0 B 0 C 0 .
Similarly, we take SSS and ASA congruence tests as axioms.
In a triangle ABC, from every vertex we can draw
— angle bisector
— median: line connecting this vertex with the midpoint of opposite side
—altitude: line from this vertex which is perpendicular to the opposite side [Existence and uniqueness
of such a line is not obvious; see below]
First theorems: supplementary and vertical angles
Theorem 1. Two different lines have at most 1 common point.
If the lines l, m do have a common point O, we say that they intersect at O. In this case, there are four
angles. A pair of angles from these four is supplementary if they share a side; vertical otherwise.
Theorem 2. (Kiselev, sections 22, 26) Supplementary angles add up to 180◦ . Vertical angles are equal.
Isosceles triangle
A triangle ABC is called isosceles if AB = AC (in this case, we call BC the base of the triangle).
Theorem 3. (Kiselev, section 35) Let 4ABC be isosceles, and let M be the midpoint of BC.
1. 4AM B ∼
= 4AM C
2. AM is also the angle bisector and the altitude
3. ∠B = ∠C
Perpendicular
Two intersecting lines are called perpendicular if one of the four angles formed by these lines is 90◦ (which
implies that the remaining three are also 90◦ ).
Theorem 4. Given a line l and a point A, there is a unique line m through A which is perpendicular to l.
Problems
• p. 14, #26, #27
• p. 19, #38, #39
• pp. 29-30, #67, #69, #74 (see definition of a kite in #64).
• Let 4ABC be such that all sides have equal length. Prove that then ∠A = ∠B = ∠C. [Such a
triangle is called equilateral.]
• Let ABCD be a quadrilateral such that AB = BC = CD = AD (such a quadilateral is called
rhombus). Let M be the intersection point of AC and BD.
1. Show that 4ABC ∼
= 4ADC
2. Show that 4AM B ∼
= 4AM D
3. Show that the diagonals are perpendicular and that the point M is the midpoint of each of the
diagonals.