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Honors Geometry Section 3.5 Triangle Sum Theorem A triangle is the figure formed by three line segments joining three noncollinear points. A B ABC C Triangles are classified according to their angles and sides. Angle classification: equiangular 3 congruent angles acute 3 acute angles right 1 right angle obtuse 1 obtuse angle Side classification: equilateral 3 congruent sides isosceles 2 congruent sides scalene 0 congruent sides Examples: Classify each triangle according to its angles and sides. right isosceles acute scalene 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 Postulate Given a line and a point not on the line, there is exactly one line through the point parallel to the given line. 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 through the given point parallel to the given line. Theorem 3.5.2: Triangle Sum Theorem The sum of the measures of the three angles of a triangle is 180 ____. (You will be asked to complete the proof of the Triangle Sum Theorem in the homework.) Examples: Find the value of x. 35 118 x 180 153 x 180 x 27 Examples: Find the value of x. 180 65 71 44 180 44 136 Examples: Find the value of x. 90 x 16 x 180 x 16 x 90 2 x 106 x 53 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 side of the triangle. Note that the exterior angle will form a _________with an interior angle of linear pair the triangle. In example b) we found x to equal 136. Note that ____________. 65 71 136 This work leads us to the following theorem. Theorem 3.5.3 Exterior Angle Theorem The 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, m1 m2 m3 Example: Find the value of x. 2 x 10 x 65 x 55 A corollary is a statement easily proven using a particular theorem. Example c) on the previous page illustrates the following corollary: 90 x 16 x 180 x 16 x 90 Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary _____________.