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CHAPTER 3: PARALLEL LINES AND PLANES Section 3-4: Angles of a Triangle TRIANGLES • • • A triangle is a figure formed by three segments that join three non-collinear points. The three non-collinear points of triangles are known as vertices (plural for vertex). The three segments that form the triangle are known as sides. TRIANGLES Triangle QRS is shown ( QRS): Q • Vertices: points Q, R, and S • Sides: QR, RS, and SQ • Angles: Q, R, and S S. R CLASSIFYING TRIANGLES Triangles can be classified by the number of congruent sides it has. Scalene Triangle Isosceles Triangle No sides congruent. At least 2 sides congruent. Equilateral Triangle All sides congruent. CLASSIFYING TRIANGLES Triangles can also be classified by their angles. • Acute (3 acute angles): • Obtuse • Right (1 obtuse angle): (1 right angle): • Equiangular (All angles congruent): AUXILIARY LINE Definition: An auxiliary line is a line, ray, or segment, that is added to a diagram to help derive information for proofs. We will use an auxiliary line in the proof for Theorem 3-11. THEOREM 3-11 PROOF D Given: 4 ABC Prove: m 1+m B A 2+m 1 3 = 180 1. Through B draw BD parallel to AC. 2. m DBC + m 5 = 180; m DBC = m 4 + m 2 3. m 4 + m 2 + m 5 = 180 4. m 4 = m 1; m 5=m 3 5. m 1 + m 2 + m 3 = 180 2 5 3 C 1. Theorem 3-8 2. Angle Addition Post. 3. 4. Substitution Property Theorem 3-2 5. Substitution Property THEOREM 3-11 & Corollaries Theorem 3-11: The sum of the measures of the angles of a triangle is 180. Corollary 1: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. 3-11 COROLLARIES Corollary 2: Each angle of an equiangular triangle has measure 60. Corollary 3: In a triangle, there can be at most one right angle or obtuse angle. Corollary 4: The acute angles of a right triangle are complementary. EXAMPLES State whether a triangle is acute, right, or obtuse if it has angles with measures: 1. 2. 3. 4. 5. 55, 43 47, 43 80, 75 33, 33 24, 66 1. 2. 3. 4. 5. Acute Right Acute Obtuse Right PRACTICE Find the m A: 1. 1. 80 50 80 A 50 2. 2. 60 40 A 60 3. 40 3. 40 80 120 A THEOREM 3-12 Theorem 3-12: The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles. 50 Remote Interior Angles Exterior Angle 120 60 70 CLASSWORK/HOMEWORK • CW: Pg. 96, Classroom Exercises 1-15 • HW: Pg. 97, Written Exercises 2-18 even