2. Warm-Up And speaking of the final frontier, what do you think is the shape of space?

3. 4.1 Apply Triangle Sum Properties Objectives: 1.To classify triangles by sides and angles 2.To find the measures of the interior and exterior angles of a triangle

4. Polygons polygon sides vertices 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.

5. 3-D Rendering 3-D rendering in digital graphics is based upon polygons.

6. Vocabulary ScaleneIsosceles Equilateral In your notebook, 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. AcuteRight ObtuseEquiangular Classified by Sides Classified by Angles

7. Types of Triangles

8. Triangles in Architecture

10. Interior vs. Exterior Angles interior exterior 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.

11. Investigation 1 Click on the button to investigate the Triangle Sum Theorem.

12. Investigation 1 1.On a clean sheet of paper, draw a large acute triangle. 2.Measure the three angles of the triangle as accurately as possible with your protractor.

13. Investigation 1 3.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.

14. Investigation 1 4.Write letters a, b, and c in the interiors of the three angles of the triangle, and carefully cut out the triangle. 5.Tear off the three angles.

15. Investigation 1 6.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?

16. Triangle Sum Theorem The sum of the measures --?--. Click the arrow buttons to watch the Flash animation of the Triangle Sum Theorem. Write and complete this statement in your notebook

17. Example 1 In  ABC below, find the measures of  1 and  2. <1 = 80 <2 = 100

18. Example 2 Prove the Triangle Sum Theorem. Given:  ABC Prove: m  a + m  b + m  c = 180° HINT: Uses parallel lines theorem and linear angle sum TRY THIS in your notebook!

19. 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

20. Big Unanswered Question So what shape is the universe? Is it basically flat, or is it curved?

21. Big Unanswered Question So what shape is the universe? Is it basically flat, or is it curved?

22. Example 3 In the right triangle below, what is the value of x + y ? 45

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

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

25. Investigation 2 1.On patty paper, draw a scalene ΔABC. 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.

26. Investigation 2 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.

27. Investigation 2 3.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.

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

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

30. Example 4 Find m<JKL. m<JKL = 35

31. Example 5 Find the values of a, b, and c. <a = 65 <b = 60 <c = 55

32. Exercise 6: SAT What is the value of c ? 160

33. Example 7 Rewrite the Triangle Sum Theorem and its Corollary in terms of radians. Do this in your notebook

34. Example 8 Find the measure of each angle in radians. <A & <C = 2π 7 <B = 3π 7

35. Assignment P. 221-224: 12, 15, 17, 19, 27, 28, 32- 39, 48-50, 52-55 Challenge Problems