Angles of a Triangle
April 14, 2008
What is a triangle?
• A triangle is the figured form
by three segments joining
three noncollinear points.
• Each of the three points is
called a vertex of the triangle
(the plural of vertex is
vertices.)
• The segments are the sides of
the triangle.
Classifying triangles by their
sides
• We often classify
triangles by the number
of congruent sides it has.
• Scalene triangle: No
sides congruent
• Isosceles triangle: At
least two sides congruent
• Equilateral triangle: All
sides congruent
•
•
•
•
Classifying triangles by their
angles
Acute: three acute angles
Obtuse: one obtuse angle
Right: one right angle
equiangular: all angles
are congruent
Theorem: The sum of the measures
of the angles of a triangle is 180.
• Part of the proof of this theorem involves
the use of an auxiliary line. An auxiliary
line is a line (or segment or ray) added to a
diagram to help in a proof.
Corollaries
• A corollary of a theorem is a statement that
can be proved easily by applying a theorem.
• It can be easily proved from the theorem
itself by simply adding a few more steps.
Corollary 1
• If two angles of one triangle are congruent
to two angles of another triangle,
then the third angles are congruent.
Corollary 2
• Each angle of an equiangular triangle has
measure 60.
Corollary 3
• In a triangle, there can be at most one right
angle or obtuse angle.
Corollary 4
• The acute angles of a right triangle are
complementary.
Interior and exterior angles
• When one side of a
triangle is extended, an
exterior angle is formed.
• Because an exterior angle
of a triangle is always a
supplement of the
adjacent interior angle of
the triangle, its measure
is related to the meausres
of the remaining angles
of the triangle, called the
remote interior angles
Now check out this one
• Notice that in both cases the measure of the
exterior angle is equal to the sum of the two
remote interior angles.
Another theorem
• The measure of an exterior angle of a
triangle equals the sum of the measures of
the two remote interior angles.