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CHAPTER 3: PARALLEL LINES AND PLANES

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 ).

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CHAPTER 3: PARALLEL LINES AND PLANES

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  1. CHAPTER 3: PARALLEL LINES AND PLANES Section 3-4: Angles of a Triangle

  2. 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.

  3. TRIANGLES Triangle QRS is shown ( QRS): • Vertices: points Q, R, and S • Sides: QR, RS, and SQ • Angles: Q, R, and S. Q S R

  4. CLASSIFYING TRIANGLES Triangles can be classified by the number of congruent sides it has. Scalene Isosceles Equilateral TriangleTriangleTriangle No sides congruent. At least 2 sides All sides congruent. congruent.

  5. CLASSIFYING TRIANGLES Triangles can also be classified by their angles. • Acute (3 acute angles): • Obtuse (1 obtuse angle): • Right (1 right angle): • Equiangular (All angles congruent):

  6. 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.

  7. Through B draw BD parallel to AC. m DBC + m 5 = 180; m DBC = m 4 + m 2 m 4 + m 2 + m 5 = 180 m 4 = m 1; m 5 = m 3 5. m 1 + m 2 + m 3 = 180 Theorem 3-8 Angle Addition Post. Substitution Property Theorem 3-2 Substitution Property THEOREM 3-11 PROOF D B Given: ABC Prove: m 1 + m 2 + m 3 = 180 4 2 5 1 3 A C

  8. 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.

  9. 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.

  10. 55, 43 47, 43 80, 75 33, 33 24, 66 Acute Right Acute Obtuse Right EXAMPLES State whether a triangle is acute, right, or obtuse if it has angles with measures:

  11. 80 60 40 PRACTICE Find the m A: 50 80 A 50 40 A 40 60 80 120 A

  12. 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. Remote Interior Angles 50 Exterior Angle 120 60 70

  13. CLASSWORK/HOMEWORK • CW: Pg. 96, Classroom Exercises 1-15 • HW: Pg. 97, Written Exercises 2-18 even

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