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Transcript
Lesson 3-1 Triangle Fundamentals Lesson 3-1: Triangle Fundamentals 1 Naming Triangles Triangles are named by using its vertices. For example, we can call the following triangle: ∆ABC ∆ACB ∆BAC ∆BCA ∆CAB ∆CBA B C A Lesson 3-1: Triangle Fundamentals 2 Opposite Sides and Angles Opposite Sides: A Side opposite to A : BC Side opposite to B : AC Side opposite to C : AB B C Opposite Angles: Angle opposite to BC : A Angle opposite to AC : B Angle opposite to AB : C Lesson 3-1: Triangle Fundamentals 3 Classifying Triangles by Sides Scalene: A triangle in which all 3 sides are different lengths. A A B C B BC = 3.55 cm C BC = 5.16 cm Isosceles: A triangle in which at least 2 sides are equal. G Equilateral: A triangle in which all 3 sides are equal. GH = 3.70 cm H Lesson 3-1: Triangle Fundamentals HI = 3.70 cm 4 I Classifying Triangles by Angles Acute: A triangle in which all 3 angles are less than 90˚. G 76 57 47 H Obtuse: I A A triangle in which one and only one angle is greater than 90˚& less than 180˚ 44 28 108 C B Lesson 3-1: Triangle Fundamentals 5 Classifying Triangles by Angles Right: A triangle in which one and only one angle is 90˚ A 56 B 90 34 C Equiangular: A triangle in which all 3 angles are the same measure. B 60 A 60 Lesson 3-1: Triangle Fundamentals 60 C 6 Classification by Sides with Flow Charts & Venn Diagrams polygons Polygon triangles Triangle scalene Scalene Isosceles isosceles equilateral Equilateral Lesson 3-1: Triangle Fundamentals 7 Classification by Angles with Flow Charts & Venn Diagrams Polygon polygons triangles Triangle right acute Right Obtuse Acute Equiangular Lesson 3-1: Triangle Fundamentals equiangular obtuse 8 Theorems & Corollaries Triangle Sum Theorem: The sum of the interior angles in a triangle is 180˚. Third Angle Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. Corollary 1: Each angle in an equiangular triangle is 60˚. Corollary 2: Acute angles in a right triangle are complementary. Corollary 3: There can be at most one right or obtuse angle in a triangle. Lesson 3-1: Triangle Fundamentals 9 Exterior Angle Theorem The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Remote Interior Angles Exterior Angle mACD mA mB Example: Find the mA. B x A (3x-22) D C D B 3x - 22 = x + 80 80 A 3x – x = 80 + 22 C mA = x = 51° 2x = 102 Lesson 3-1: Triangle Fundamentals 10 Median - Special Segment of Triangle Definition: A segment from the vertex of the triangle to the midpoint of the opposite side. B Since there are three vertices, there are three medians. C A F E D In the figure C, E and F are the midpoints of the sides of the triangle. DC , AF , BE are the medians of the triangle. Lesson 3-1: Triangle Fundamentals 11 Altitude - Special Segment of Triangle Definition: The perpendicular segment from a vertex of the triangle B to the segment that contains the opposite side. C AF , BE , DC are the altitudes of the triangle. B In a right triangle, two of the altitudes of are the legs of the triangle. B I A K A D F E A AB, AD, AF altitudes of right F F D In an obtuse triangle, two of the altitudes are outside of the triangle. BI , DK , AF altitudes of obtuse Lesson 3-1: Triangle Fundamentals 12 D Perpendicular Bisector – Special Segment of a triangle Definition: A line (or ray or segment) that is perpendicular to a segment at its midpoint. The perpendicular bisector does not have to start from a vertex! P Example: M A E A C D B In the scalene ∆CDE, AB is the perpendicular bisector. B O L N In the right ∆MLN, AB is the perpendicular bisector. Lesson 3-1: Triangle Fundamentals R In the isosceles ∆POQ, PR is the perpendicular bisector. 13 Q