Section 2.4 The Angles of a Triangle Def: A triangle (△)is the union of three segments that are determined by three noncolinear points. Terminology: Vertices: The noncolinear points Sides: Lines connecting the vertices Naming Convention: Named by the three points Points are named in alphabetical order 7/19/2019 Section 2.4 Nack

Triangles Classified by Congruent Sides p. 93 7/19/2019 Section 2.4 Nack

Triangles Classified by Angles p. 93 7/19/2019 Section 2.4 Nack

Sum of the Measures of Interior Angles Theorem 2.4.1: In a triangle, the sum of the measures of the interior angles is 180. Picture Proof Given: ΔABC Prove: mA + mB + mC = 180 Proof: ↔ _ Through C, draw line ED || AB. (auxillary line used to facilitate proof) We see that: mA + mB + mC = 180. But m1 = mA and m3 = mB (alternate interior angles are equal) Then mA + mB + mC = 180 7/19/2019 Section 2.4 Nack

Corollaries to Theorem 2.4.1 Corollaries: Theorems that follow directly from a previous theorem. Corollary 2.4.2: Each angle of an equiangular triangle measures 60. Corollary 2.4.3: The acute angles of a right triangle are complementary. Ex. 3 p. 95 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. Ex. 4 p. 95 Corollary 2.4.5:The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles. Fig. 2.26, Ex. 5 p. 96 7/19/2019 Section 2.4 Nack

Determined: Only one possible figure can be drawn Defining Auxillary Figures for Proofs Determined, Undertermined and Overdetermined Determined: Only one possible figure can be drawn Underdetermined: Vague so that many figures could satisfy the condition. Overdetermined: There is no figure that satisfies the conditions. Ex: Draw a line that is parallel and perpendicular to the same line. Draw an equilateral triangle Draw a line through a point perpendicular to a given line. 7/19/2019 Section 2.4 Nack