Download The sum of the interior angles of a triangle makes

A proof that can be
proved in Euclidean
geometry, yet not in
Non-Euclidean
geometry.
The sum of the interior angles
of a triangle makes two right
angles
All of these
triangles have
interior angles
summing to two
right angles, or
180 degrees, or
π radians.
For any triangle, the sum of the interior
angles is always equal to two right
angles.
• We will prove that
angle ABC plus angle
BCA plus angle CAB
is equal to two right
angles.
• Draw CE,
parallel to AB.
• Extend line BC
to a point D.
• CAB = ACE
 If two parallel lines are
cut by a transversal,
the alternate interior
angles are equal.
• ABC = ECD
 If two parallel lines are
cut by a transversal,
the corresponding
angles are equal.
• ABC + CAB =
ACE + ECD
 Equals added to equals are
equal.
• ABC + CAB = ACD
 ACE + ECD= ACD
ABC + CAB + BCA =
ACD + BCA
 Add BCA. Equals added
to equals are equal.
ACD + BCA =
Two right angles
 Two angles on a straight line
are either two right angles,
or equal to two right angles.
ABC+CAB + BCA is
equal to two right
angles.
 Things which equal the
same thing are also equal
to one another.
Why can’t this
theorem be
proved in
Non-Euclidean
geometry?
The sum of the interior
angles of a triangle
equals two right triangles.
If two parallel lines are cut by a
transversal, the alternate interior
angles are equal.
If two parallel lines are cut by a
transversal, the corresponding
angles are equal.
The key is in the
justification of these two
steps in our proof. We
need to use the parallel
postulate to prove each
of these statements.
Parallel Postulate: If a straight line falling on
two straight lines makes the interior angles on
the same side less than two right angles, the
two straight lines, if produced indefinitely,
meet on that side on which are the angles
less than the two right angles.
If  AEF +  CFE is less than two right angles,
lines AB and CD will surely meet.
By denying the parallel
postulate, we can
create many new
(Non Euclidean)
geometries.
• This triangle, on
the hyperbolic
plane, has
interior angles
adding to one
right angle.
• This triangle, on
the elliptic plane,
has interior
angles summing
to 232 degrees.
In this
diagram, the
line AB has
two parallel
lines passing
through the
point C.
References
T. L. Heath (1956). Euclid’s Elements. New
York: Dover.
• Illustrations created with Cinderella.