Honors Geometry Section 3.5 Triangle Sum Theorem

1. Honors Geometry Section 3.5Triangle Sum Theorem

2. A triangle is the figure formed by three line segments joining three noncollinear points. A B C

3. Triangles are classified according to their angles and sides.Angle classification: Side classification:equiangularequilateralacuteisoscelesrightscaleneobtuse 3 congruent angles 3 congruent sides 2 congruent sides 3 acute angles 1 right angle 0 congruent sides 1 obtuse angle

4. Examples: Classify each triangle according to its angles and sides.

5. In Unit III, we discussed parallel lines in rather great detail. We did, however, fail to discuss Euclid’s Parallel Postulate. Let’s remedy that now.Theorem 3.5.1 The Parallel PostulateGiven a line and a point not on the line, there is exactly one line through the point parallel to the given line.

6. As with many of the postulates that we have discussed thus far, this may seem obvious but it plays a very important role in Euclidean geometry. In Euclidean geometry, planes are flat, but there are other ways of thinking of a plane. In spherical geometry planes are the surface of a sphere (i.e. a globe) and lines are great circles (i.e. the equator or any of the lines of longitude). In spherical geometry the Parallel Postulate would read“given a line and a point not on the line there are no lines throughthe given point parallel to the given line.

7. Theorem 3.5.2: Triangle Sum TheoremThe sum of the measures of the three angles of a triangle is ____. (You will be asked to complete the proof of the Triangle Sum Theorem in the homework.)

8. Examples: Find the value of x.

9. Examples: Find the value of x.

10. Examples: Find the value of x.

11. The angle of x° in example b) is called an exterior angle of the triangle. An exterior angle of a triangle is formed by extending a sideof the triangle.Note that the exterior angle will form a _________with an interior angle of the triangle. linear pair

12. In example b) we found x to equal 136. Note that ____________. This work leads us to the following theorem.

13. Theorem 3.5.3 Exterior Angle TheoremThe measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles.For the triangle to the right,

14. Example: Find the value of x.

15. A corollary is a statement easily proven using a particular theorem.

16. Example c) on the previous page illustrates the following corollary:Corollary to the Triangle Sum TheoremThe acute angles of a right triangle are _____________. complementary