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Lesson 3-4 Angles of a Triangle (page 93) Essential Question How can you apply parallel lines (planes) to make deductions? Angles of a Triangle . A . B .C TRIANGLE: The figure formed by 3 segments joining 3 . A B . . ∆ ACB noncollinear points. ∆ BAC ∆ BCA ∆ ABC Symbol: ____________ ∆ CAB three points. VERTEX: each of the ______ ∆ CBA C Vertices: A _____ B _____ C _____ Sides: _____ _____ _____ Angles: ∠A ∠B _____ _____∠C _____ AB BC AC Classifications of Triangles by Sides Scalene Triangle No sides are congruent . Classifications of Triangles by Sides Isosceles Triangle At least two sides are congruent . Classifications of Triangles by Sides Equilateral Triangle All sides are congruent . Classifications of Triangles by Angles Acute Triangle Three acute angles. Classifications of Triangles by Angles Right Triangle One right (90º) angle. Classifications of Triangles by Angles Obtuse Triangle One obtuse angle. Classifications of Triangles by Angles Equiangular Triangle All angles are congruent . Auxiliary Line: A line (ray or segment) added to a diagram to help in a proof . Please note that this is to HELP in a proof. This does not give you license to add lines to every diagram. There are times when this may be done, but please BEWARE! Theorem 3-11 The sum of the measures of the angles of a triangle is 180º . B Given: ∆ ABC Prove: m∠1 + m∠2 + m∠3 = 180º 1 A 2 3 C D Given: ∆ ABC B 4 Prove: m∠1 + m∠ 2 + m∠3 = 180º 2 5 See page 94! Proof: Statements A 1 3 Reasons C 1. Through B draw line BD parallel to line AC Through a point outside a line, there is exactly one line parallel to the given line. 2. m∠ DBC +m∠5 = 180º __________________________________________ ∠ - Addition Postulate _____________________________________________ 3. m∠ DBC = m∠2 + m∠4 m∠2 + m∠4 + m∠5 = 180º __________________________________________ 4. ∠1 ∠4 OR m∠1 = m∠4 __________________________________________ _____________________________________________ 5. m∠1 + m∠2 + m∠ 3 = 180º __________________________________________ _____________________________________________ ∠3 ∠5 OR m∠3 = m∠ 5 Substitution Property _____________________________________________ || - lines ⇒ AIA Substitution Property Example # 1. Find the value of “x”. xº x + 40 + 90 = 180 x + 130 = 180 50 x = _____ 40º Example # 2. Find the value of “x”. x + 100 + 35 = 180 xº x + 135 = 180 45 x = _____ 100º 35º Example # 3. Find the value of “x”. x + x + 70 = 180 xº 2x + 70 = 180 2x = 110 70º xº 55 x = _____ COROLLARY: A statement that can be proved easily by applying a theorem. Corollary 1 If two angles of one triangle are congruent to two angles of another triangle, congruent . then the third angles are A B X Y Z C If ∠A ∠X and ∠B ∠Y, then ∠C ∠Z Corollary 2 Each angle of an equiangular triangle has a measure 60º . xº xº xº If 3 x = 180, then x = 60. Corollary 3 In a triangle, there can be at most one right angle or obtuse angle. Sum = 90º Sum < 90º m > 90º Corollary 4 The acute angles of a right triangle are complementary . Sum = 90º If 2 ∠’s sum = 90º, then they are complementary. Example # 4. Find the value of “x”. 40º 50º xº 40º 50 x = _____ Example # 5. Find the value of “x”. xº xº xº 60 x = _____ EXTERIOR ANGLE: (of a triangle) the angle formed when one side of a triangle is extended . Example: ∠4 4 3 Example: 2 ∠1 & ∠2 1 REMOTE INTERIOR ANGLES: the angles of the triangle not adjacent to the exterior angle. EXTERIOR ANGLE: Example: ∠4 1 4 2 3 REMOTE INTERIOR ANGLES: Example: ∠1 & ∠2 Theorem 3-12 The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles. B 2 1 A Given: ∆ ABC Prove: m∠1 + m∠2 = m∠ 4 3 4 C Given: ∆ ABC B Prove: m∠1 + m∠2 = m∠4 2 See page 96 C.E. #15! 1 Proof: A 3 4 C 1. Statements m∠1 + m∠2+ m∠ 3 = 180º __________________________________________ Reasons Sum of m. of∠‘s of ∆ = 180º _____________________________________________ 2. m∠3 + m∠ 4 = 180º __________________________________________ _____________________________________________ 3. m∠1+ m∠2 + m∠3 = m∠3 + m∠4 __________________________________________ _____________________________________________ 4. m∠3 = m∠3 __________________________________________ _____________________________________________ 5. m∠1 + m∠ 2 = m∠4 __________________________________________ _____________________________________________ ∠- Addition Postulate Substitution Property Reflexive Property Subtraction Property Example # 6. Find the value of “x”. 80º x + 80 = 120 40 x = _____ 120º xº Example # 7. Find the value of “x”. 3x = x + 100 2x = 100 xº 50 x = _____ 100º 3xº Example # 8. Find the value of “x”. 60º x = 90 + 60 150 x = _____ xº Assignment Written Exercises on pages 97 to 99 RECOMMENDED: 1 to 9 odd numbers REQUIRED: 10, 11, 13, 15, 17, 18, 19, 20, 25, 26 Do the Paper Triangle Proofs How can you apply parallel lines (planes) to make deductions?