1. 4.1 Triangles and Angles

2. Triangles are always classified in two ways:
Definition: Triangle
A triangle is a figure formed by three segments joining three noncollinear points.
Triangles are always classified in two ways:
By Sides
By Angles

3. 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

4. Classification by Sides
Equilateral Triangle = 3 congruent sides
Isosceles Triangle = At least 2 congruent sides
Scalene Triangle = No congruent sides

5. Classification by Angles
Acute Triangle:
Three Acute Angles (all measure less than 90o)
Equiangular Triangle: Three congruent angles (each measures 60o)
Right Triangle;
One Right Angle (90o)
Obtuse Triangle: One obtuse Angle (more than 90o) and two acute angles (less than 90o)

6. Example 1: Classifying Triangles
When you classify a triangle, you must give TWO classifications (sides & angles.)
How would you classify triangle ABC?
Sides: _____________
Angles: ____________

7. Example 2: Classifying Triangles
How would you classify this triangle?
Sides: _____________
Angles: ____________

8. Example 3: Classifying Triangles
How would you classify this triangle?
Sides:
Angles:

9. Special Labels for Right and Isosceles Triangles
Leg
Leg
Hypotenuse
Leg
Base
Leg

10. 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.

11. Triangle Sum Theorem
The three angles inside of a triangle must always add to 180o.
If m<A = 75o and m<B = 85o,
what must the measure of
angle C be?

12. 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
If m<A = 90o and m<B = 45o, then what is m<1?
If m<1 = 102o and m<B = 55o, then what is m<A? 1 A B C