1. Triangle Fundamentals
Lesson 5-1
Triangle Fundamentals

2. Naming Triangles Triangles are named by using its vertices.
For example, we can call the following triangle: A B C
∆ABC
∆ACB
∆BAC
∆BCA
∆CAB
∆CBA

3. Opposite Sides and Angles
Side opposite to A :
Side opposite to B :
Side opposite to C :
Opposite Angles:
Angle opposite to : A
Angle opposite to : B
Angle opposite to : C

4. Classifying Triangles by Sides
Scalene:
A triangle in which all 3 sides are different lengths. BC =
5.16 cm B C A BC =
3.55 cm A B C
AB = 3.47 cm
AC = 3.47 cm
AB = 3.02 cm
AC = 3.15 cm
Isosceles:
A triangle in which at least 2 sides are equal. HI =
3.70 cm G H I
Equilateral:
A triangle in which all 3 sides are equal.
GI = 3.70 cm
GH = 3.70 cm

5. Classifying Triangles by Angles
Acute:
A triangle in which all 3 angles are less than 90˚. 57 47 76 G H I
Obtuse:
108 44 28 B C A
A triangle in which one and only one angle is greater than 90˚& less than 180˚

6. Classifying Triangles by Angles
Right:
A triangle in which one and only one angle is 90˚
Equiangular:
A triangle in which all 3 angles are the same measure.

7. Theorems & Corollaries
Triangle Sum Theorem:
The sum of the interior angles in a triangle is 180˚.
Third Angle Theorem:
If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.
Corollary 1:
Each angle in an equiangular triangle is 60˚.
Corollary 2:
Acute angles in a right triangle are complementary.
There can be at most one right or obtuse angle in a triangle.
Corollary 3:

8. 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.
Remote Interior Angles A
Exterior Angle D
Example:
Find the mA. B C
3x - 22 = x + 80
3x – x =
2x = 102
mA = x = 51°

9. Perpendicular Bisector – Special Segment of a triangle
A line (or ray or segment) that is perpendicular to a segment at its midpoint.
Definition:
The perpendicular bisector does not have to start from a vertex! R O Q P
Example: M L N C D A E A B B
In the isosceles ∆POQ, is the perpendicular bisector.
In the scalene ∆CDE, is the perpendicular bisector.
In the right ∆MLN, is the perpendicular bisector.

10. Median - Special Segment of Triangle
Definition:
A segment from the vertex of the triangle to the midpoint of the opposite side. B A D E C F
Since there are three vertices, there are three medians.
In the figure C, E and F are the midpoints of the sides of the triangle.

11. Altitude - Special Segment of Triangle
The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side.
Definition: B A D F
In a right triangle, two of the altitudes of are the legs of the triangle. B A D F I K
In an obtuse triangle, two of the altitudes are outside of the triangle.