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4.1 Triangles and Angles
Definition: Triangle
A triangle is a figure formed by three
segments joining three noncollinear points.
Triangles are always classified in two ways:
1. By Sides
2. By Angles
Subtending Angles & Sides
• When two sides are equal then their subtending angles
must be equal.
• Likewise, when two angles are equal then their
subtending sides must be equal.
B
Angle A subtends Side BC
Angle B subtends Side AC
Angle C subtends Side AB
If AB = BC, then which two
angles must be equal?
If <B = <C, then which two sides
must be equal?
A
C
Classification by Sides
Equilateral Triangle = 3 congruent sides
Isosceles Triangle = At least 2 congruent sides
Scalene Triangle = No congruent sides
Classification by Angles
Acute Triangle:
Three Acute Angles (all
measure less than 90o)
Right Triangle;
One Right Angle (90o)
Equiangular Triangle:
Three congruent angles (each
measures 60o)
Obtuse Triangle: One
obtuse Angle (more than 90o)
and two acute angles (less
than 90o)
Example 1: Classifying Triangles
When you classify a triangle, you must give
TWO classifications (sides & angles.)
How would you classify
triangle ABC?
Sides: _____________
Angles: ____________
Example 2: Classifying Triangles
How would you classify this triangle?
Sides: _____________
Angles: ____________
Example 3: Classifying Triangles
How would you classify
this triangle?
Sides:
Angles:
Special Labels for Right and Isosceles Triangles
Hypotenuse
Leg
Leg
Leg
Base
Leg
By extending the sides we create
Interior and Exterior Angles
Interior Angles are “inside” the triangle. Exterior Angles are “outside
the triangle. Label each interior angle with an “I” and each exterior
angle with an “E.” When an interior angle and an exterior angle are
together they make a straight line and sum to 180o.
Triangle Sum Theorem
• The three angles inside of a triangle must
always add to 180o.
mA  mB  mC  180
If m<A = 75o and m<B = 85o,
what must the measure of
angle C be?
A special and useful property!
• The measure of an exterior angle of a triangle is equal to
the sum of the measures of the 2 nonadjacent interior
angles
m1  mA  mB
If m<A = 90o and m<B = 45o, then what is m<1?
B
If m<1 = 102o and m<B = 55o, then what is m<A?
1
A
C