1. Triangle Fundamentals
Lesson 3-1
Triangle
Fundamentals

2. For example, we can call the following triangle:
Classifying Triangles
We name a triangle using its vertices.
For example, we can call the following triangle:
∆ABC
∆ACB
∆BAC
∆BCA
∆CAB
∆CBA

3. Opposite Sides and Angles
We say that is opposite .

4. Triangles can be classified by:
Their sides
Scalene
Isosceles
Equilateral
Their angles
Acute
Right
Obtuse
Equiangular

5. A triangle in which all 3 sides are different lengths
Scalene Triangle
A triangle in which all 3 sides are different lengths

6. Isosceles Triangle
A triangle in which at least 2 sides are equal

7. A triangle in which all 3 sides are equal
Equilateral Triangle
A triangle in which all 3 sides are equal

8. Acute Triangle
A triangle in which all 3 angles are less than 90˚

9. Right Triangle
A triangle in which exactly one angle is 90˚

10. Obtuse Triangle
A triangle in which exactly one angle is greater than 90˚and less than 180˚

11. A triangle in which all 3 angles are the same measure
Equiangular Triangle
A triangle in which all 3 angles are the same measure

12. Classification of Triangles Flow Charts Venn Diagrams
with
Flow Charts
and
Venn Diagrams

13. Classification by Sides
polygons
Polygon
triangles
Triangle
scalene
isosceles
Scalene
Isosceles
equilateral
Equilateral

14. Classification by Angles
polygons
Polygon
triangles
Triangle
right
acute
equiangular
Right
Obtuse
Acute
obtuse
Equiangular

15. Triangle Theorems

16. Triangle Sum Theorem
The sum of the interior angles in a triangle is 180˚.

17. Third Angle Corollary
If two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent.

18. Corollary
Each angle in an equiangular triangle is 60˚.

19. Corollary
There can be at most one right or obtuse angle in a triangle.

20. Corollary
Acute angles in a right triangle are complementary.

21. Exterior Angle Theorem
The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

22. Special Segments of Triangles

23. Introduction There are three segments associated with triangles:
Medians, Altitudes and Perpendicular Bisectors

24. Medians
Definition: a segment from a vertex to the midpoint of the opposite side
The median is in red here.

25. Medians How many medians does every triangle have?
Since there are three vertices, there are three medians.
Look at this example where D, E and F are the midpoints:

26. Altitudes
Definition: the perpendicular segment from a vertex to the line that contains the opposite side
Every triangle has three altitudes.
See the altitudes:

27. Altitudes in a right triangle
Two of the altitudes of a right triangle are also legs of the triangle.
See this example:
All of the altitudes are colored black.

28. Altitudes in an obtuse triangle
Two of the altitudes here must be outside of the triangle.
See this example:
The original triangle is solid blue and the altitudes are black.

29. Perpendicular Bisector
Definition: a line (or ray or segment) that is perpendicular to a segment at its midpoint.
The perpendicular bisector does not have to start from a vertex!
Every triangle has three perpendicular bisectors.

30. Example 1:
Draw the perpendicular bisector of in this scalene triangle.
Notice how the bisector does not originate from a vertex of the triangle.

31. Example 2: See this example in a right triangle.
Draw the perpendicular bisector of in this triangle.

32. Special Case
The perpendicular bisector can pass through a vertex in an isosceles triangle (and therefore in an equilateral triangle).
In the isosceles triangle, when the bisector is drawn to the base, it will pass through the vertex angle point.

33. Isosceles Triangle
In , if is the base, draw the perpendicular bisector to .
P is the vertex of the triangle.
The perpendicular bisector will pass through P when drawn to the base.