1. 4.1 Apply Triangle Sum Properties
Objectives:
To classify triangles by sides and angles
To find the measures of the interior and exterior angles of a triangle

2. Polygons
A closed plane figure is a polygon if it is formed by 3 or more line segments (sides), joined endpoint to endpoint (vertices) with each side intersecting exactly two others.

3. 3-D Rendering
3-D rendering in digital graphics is based upon polygons.

4. Vocabulary
As a group, define each of these types of triangles without your book. Draw a picture for each word and leave a bit of space for additions and revisions.
Scalene
Isosceles
Equilateral
Classified by Sides
Acute
Right
Obtuse
Equiangular
Classified by Angles

5. Types of Triangles

6. Triangles in Architecture

7. Triangles in Architecture

8. Interior vs. Exterior Angles
All the angles inside the triangle are called interior angles. If you extend the sides of the triangle, then the angles that form a linear pair with the interior angles are called exterior angles.

9. Investigation 1
Use the following Investigation to complete the Triangle Sum Theorem.

10. Investigation 1
On a clean sheet of paper, draw a large acute triangle.
Measure the three angles of the triangle as accurately as possible with your protractor.

11. Investigation 1
Find the sum of the measures of the three angles the triangle.
What appears to be the sum of the three angle measures in every triangle?
Let’s check the sum another way.

12. Investigation 1
Write letters a, b, and c in the interiors of the three angles of the triangle, and carefully cut out the triangle.
Tear off the three angles.

13. Investigation 1
Arrange the three angles so that their vertices meet at a point. How does this arrangement show the sum of the angle measures? What is the sum of the three angles of any triangle?

14. Triangle Sum Theorem The sum of the measures --?--.
Click the arrow buttons to watch the Flash animation of the Triangle Sum Theorem.

15. Example 1
In ABC below, find the measures of  1 and 2.

16. Example 2
Prove the Triangle Sum Theorem. Given: ABC Prove: ma + mb + mc = 180°

17. Non-Euclidean Triangles
Does the Triangle Sum Conjecture hold true in either hyperbolic or elliptic geometry? The answer may reveal secrets of the universe.
Hyperbolic
Elliptic

18. Big Unanswered Question
So what shape is the universe? Is it basically flat, or is it curved? How could The Triangle Sum Theorem help answer this question?

19. Big Unanswered Question
So what shape is the universe? Is it basically flat, or is it curved? How could The Triangle Sum Theorem help answer this question?

20. Example 3
In the right triangle below, what is the value of x + y?

21. Triangle Sum Theorem Corollary
The acute angles of a right triangle are complementary.

22. Investigation 2
Use this Investigation to discover a relationship between an exterior angle of a triangle and its two remote interior angles.

23. Investigation 2
On patty paper, draw a scalene ABC. Extend segment AB through point B and label a point D beyond point B. As shown, put an a in the interior of A, a b in the interior of B, a c in the interior of C, and an x in exterior angle CBD.

24. Investigation 2
Copy the two remote interior angles A and C onto another patty paper. Cut the patty paper into two pieces, with an angle on each piece.

25. Investigation 2
Place the two remote interior angles A and C on the exterior angle to compare the sum of their measures against x, the measure of the exterior angle.

26. Investigation 2
How does the sum of the measures of the two remote interior angles compare with the measure of the exterior angle?

27. Exterior Angle Theorem:
The measure of an exterior angle of a triangle is equal to the sum of its two remote interior angles.

28. Example 4
Find mJKL.

29. Example 5
Find the values of a, b, and c.

30. Exercise 6: SAT
What is the value of c?

31. Example 7
Rewrite the Triangle Sum Theorem and its Corollary in terms of radians.

32. Example 8
Find the measure of each angle in radians.