1. Triangle Fundamentals
Intro to G.10
Triangle
Fundamentals
Modified by Lisa Palen

2. Triangle What’s a polygon?
Definition: A triangle is a three-sided polygon.
What’s a polygon?

3. Polygons
Definition:
A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints.
These figures are not polygons
These figures are polygons

4. Definition of a Polygon
A polygon is a closed figure in a plane formed by a finite number of segments that intersect only at their endpoints.

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

6. Classifying Triangles by Sides
Scalene:
A triangle in which no sides are congruent. 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 congruent. HI =
3.70 cm G H I
Equilateral:
A triangle in which all 3 sides are congruent.
GI = 3.70 cm
GH = 3.70 cm

7. Classifying Triangles by Angles
Obtuse:
108 44 28 B C A
A triangle in which one angle is....
obtuse.
Right:
A triangle in which one angle is...
right.

8. Classifying Triangles by Angles
Acute: 57 47 76 G H I
A triangle in which all three angles are....
acute.
Equiangular:
A triangle in which all three angles are...
congruent.

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

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

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

12. Naming Triangles We name a triangle using its vertices.
For example, we can call this triangle:
∆ABC
∆ACB
Review: What is ABC?
∆BAC
∆BCA
∆CAB
∆CBA

13. Parts of Triangles Every triangle has three sides and three angles.
For example, ∆ABC has
Sides: Angles:
 CAB
 ABC
 ACB

14. Opposite Sides and Angles
Side opposite of BAC :
Side opposite of ABC :
Side opposite of ACB :
Opposite Angles:
Angle opposite of : BAC
Angle opposite of : ABC
Angle opposite of : ACB

15. Interior Angle of a Triangle
An interior angle of a triangle (or any polygon) is an angle inside the triangle (or polygon), formed by two adjacent sides.
For example, ∆ABC has interior angles:
 ABC,  BAC,  BCA

16. Exterior Angle
An exterior angle of a triangle (or any polygon) is an angle that forms a linear pair with an interior angle. They are the angles outside the polygon formed by extending a side of the triangle (or polygon) into a ray.
Exterior Angle
Interior Angles A
For example, ∆ABC has exterior angle ACD, because ACD forms a linear pair with ACB. D B C

17. Interior and Exterior Angles
The remote interior angles of a triangle (or any polygon) are the two interior angles that are “far away from” a given exterior angle. They are the angles that do not form a linear pair with a given exterior angle.
For example, ∆ABC has exterior angle:
ACD and
remote interior angles A and B
Exterior Angle
Remote Interior Angles A D B C

18. Triangle Theorems

19. m<A + m<B + m<C = 180
Triangle Sum Theorem
The sum of the measures of the interior angles in a triangle is 180˚.
m<A + m<B + m<C = 180
IGO GeoGebra Applet

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

21. Third Angle Corollary Proof
Given:
The diagram
Prove:
C  F
statements
reasons
1. A  D, B  E
2. mA = mD, mB = mE
3. mA + mB + m C = 180º
mD + mE + m F = 180º
4. m C = 180º – m A – mB
m F = 180º – m D – mE
5. m C = 180º – m D – mE
6. mC = mF
7. C  F
1. Given
2. Definition: congruence
3. Triangle Sum Theorem
Subtraction Property of Equality
Property: Substitution
Definition: congruence
QED

22. Corollary Each angle in an equiangular triangle measures 60˚. 60 60

23. Corollary
There can be at most one right or obtuse angle in a triangle.
Example
Triangles???

24. Corollary
Acute angles in a right triangle are complementary.
Example

25. 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
x = 51
mA = x = 51°

26. Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
GeoGebra Applet (Theorem 1)

27. Special Segments of Triangles

28. Introduction There are four segments associated with triangles:
Medians
Altitudes
Perpendicular Bisectors
Angle Bisectors

29. 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.

30. 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 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.

31. 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.