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Section 2.4 The Angles of a Triangle

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

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Section 2.4 The Angles of a Triangle

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  1. 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 Section 2.4 Nack

  2. Triangles Classified by Congruent Sidesp. 93 Section 2.4 Nack

  3. Triangles Classified by Anglesp. 93 Section 2.4 Nack

  4. 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 Section 2.4 Nack

  5. 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 Section 2.4 Nack

  6. Defining Auxillary Figures for ProofsDetermined, 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. Section 2.4 Nack

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