ASSIGNMENT 6 – EUCLIDEAN GEOMETRY I
Angles, lines, and parallels
Undefined terms: point, line
Defined terms and Theorems:
 Segment: A segment AB consists of points A and B and all those points C on the line
through A and B such that C is between A and B (The term ‘between’ in the definition is
undefined. We will assume that ‘betweenness’ works in the way our intuition tells us it
should. The term ‘on’ is also undefined).
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 Ray: A ray AB consists of the segment AB together with those points C on the line
through A and B such that B is between A and C.
 Angle with vertex A: The angle with vertex A consists of the point A together with
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two rays AB and AC ( called the sides of the angle). We denote the angle by BAC or
CAB .
 Vertical angles: Two angles with a common vertex and whose sides form two lines
are called vertical angles.
 Supplementary: Two angles are supplementary if they share a common side and the
other sides lie in opposite directions on the same line.
We assume:
1. The measure of an angle is a real number between 0 and 180 degrees
2. Congruent angles have the same angle measure
3. The measure of right angle is 90 degrees
4. If two angles are supplementary, then the sum of the measures of the angles is
180 degrees.
 Degree measure: The degree measure of ABC will be denoted by mABC .
We assume: For any segment AB we have a way of associating a real number, called the
length of AB , satisfying the usual notions of Euclidean length for example congruent
segments have the same length.
 The length of a segment AB will be denoted by AB
 A right angle is an angle that has a supplementary angle to which it is congruent.
 Two lines that intersect are perpendicular if one of the angles made at the
intersection is a right angle.
 Two lines are parallel if they do not intersect.
Exercise: Define “bisector of a segment’ and ‘bisector of an angle’.
Theorems
1. Given an angle or segment, the bisector of that angle or segment can be constructed.
2. Given a line and a point either on or off the line, the perpendicular to the line through
the point can be constructed.
3. If two straight lines cross one another they make the vertical angles congruent. Also,
the angles formed by one of the lines crossing the other will have angle measures adding
to two right angles.
4. If the measures of two adjacent angles add to 180 degrees, then the angles form a line.
5. Given an angle, a line (or part of the line), and a point on the line, an angle whose
vertex is at the point, and whose measure is equal to the measure of the given angle, can
be constructed.
Exercise: Let a line n cross two lines l and m. Use the figure to define “interior angles’,
‘alternate angles’, ‘exterior angles’, and ‘corresponding angles’.
 We will say that an angle is greater than another angle if its angle measure is greater
than the other angle’s measure.
Theorems
6. (The Exterior Angle Theorem) Given a triangle, if one of the sides is extended, then
the exterior angle produced is greater than either of the two interior and opposite angles.
7. If a line n falling on two lines l and m makes the alternate interior angles congruent
to one another, then the two lines are parallel.
8. If a line n falling on two lines l and m makes corresponding angles congruent, or if
the sum of the measures of the interior angles on the same side equal two right angles,
then the two lines are parallel.
(NOTE: All the preceding theorems are independent of the 5th Postulate – such
results form the basis of what is called absolute or neutral geometry)
9. If a line n falling on two parallel lines l and m, then alternate interior angles are
congruent, corresponding angles are congruent, and the sum of the measures of the
interior angles on the same side of n equal two right angles.
NOTE: In doing the exercises below, use only the facts given in the definitions. Do not
impose preconceived notions on the terms.
Exercises:
1. Use Playfair’s postulate to prove Theorem 9.
2. Use the Exterior Angle Theorem to show that the sum of the measures of two interior
angles of a triangle is always less than 180 degrees.
3. Each of the following statements are about a specific term of this section. Determine,
solely on the basis of the definition of that term, if the statement is true or false:
a) A right angle is angle whose measure is 90 degrees.
b) An angle is the set of points lying between two rays that have a common vertex.
c) An exterior angle results from a line crossing two other lines.
d) A line is the union of two opposite rays
4. Use previously defined terms to define the following. Do not use imprecise and
colloquial language in your definitions:
a) Midpoint of a segment
b) Perpendicular bisector of a segment
c) Triangle defined by three non-collinear points
d) Equilateral triangle
5. Show that Euclid’s fifth postulate implies Playfair’s Postulate.
6. Prove the converse of Exercise 5. above.