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
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2. Triangles Classified by Congruent Sides p. 93
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3. Triangles Classified by Angles p. 93
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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
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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
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6. 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.
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