1. Math 2 Geometry Based on Elementary Geometry, 3 rd ed, by Alexander & Koeberlein 2.4 The Angles of a Triangle

2. Definition A triangle is the union of three line segments that are determined by three non-collinear points.

3. Definition A triangle is the union of three line segments that are determined by three non-collinear points. A B C

4. Definition A triangle is the union of three line segments that are determined by three non-collinear points. A B C

5. Definition A triangle is the union of three line segments that are determined by three non-collinear points. A B C  ABC

6. Informal Definitions Each point A, B, C is a vertex. (pl. Vertices) Segments AB, BC, and CB are sides. Point D is in the interior of the triangle. Point E is in the exterior of the triangle. A B C

7. Triangles classified by sides ScaleneNo congruent sides IsoscelesAt least two  sides EquilateralThree  sides

8. Triangles classified by angles AcuteAll angles acute ObtuseOne obtuse angle RightOne right angle EquiangularAll angles 

9. Theorem 2.4.1 In a triangle, the sum of the measure of the interior angle is 180 . What is an interior angle? We can prove this theorem A B C

10. If the sum of the measures of the interior angles of a triangle is 180  … … what can we say about the angles in an equiangular triangle?

11. Corollary 2.4.1 Each angle of an equiangular triangle measures 60 

12. If the sum of the measures of the interior angles of a triangle is 180  … … what can we say about the angles in a right triangle? A CB

13. Corollary 2.4.3 The acute angles of a right triangle are complementary. Are the two non-right angles really acute? m  A + m  B + m  C=180 A CB

14. Corollary 2.4.4 If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

15. Corollary 2.4.4 If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. Notice that this does not mean the two triangles are necessarily congruent.

16. Corollary 2.4.5 The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles. 13 4 2