* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Classifying Triangles
Survey
Document related concepts
Penrose tiling wikipedia , lookup
Tessellation wikipedia , lookup
Dessin d'enfant wikipedia , lookup
Multilateration wikipedia , lookup
Golden ratio wikipedia , lookup
Apollonian network wikipedia , lookup
Euler angles wikipedia , lookup
Rational trigonometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euclidean geometry wikipedia , lookup
Incircle and excircles of a triangle wikipedia , lookup
Transcript
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 Opposite Sides and Angles Opposite Sides: A Side opposite to A : BC Side opposite to B : AC Side opposite to C : AB Opposite Angles: Angle opposite to BC : A Angle opposite to AC : B Angle opposite to AB : C B C Classifying Triangles by Sides Scalene: A triangle in which all 3 sides are different lengths. A A B C BC = 3.55 cm B C BC = 5.16 cm Isosceles: A triangle in which at least 2 sides are equal. Equilateral: A triangle in which all 3 sides are equal. G GH = 3.70 cm H HI = 3.70 cm 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 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 60 C Classification by Sides with Flow Charts & Venn Diagrams polygons Polygon triangles Triangle scalene Scalene Isosceles isosceles equilateral Equilateral Classification by Angles with Flow Charts & Venn Diagrams Polygon polygons triangles Triangle right acute Right Obtuse Acute Equiangular equiangular obtuse 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. 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 3x - 22 = x + 80 80 x A (3x-22) D C 3x – x = 80 + 22 2x = 102 A D B C mA = x = 51° 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. 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. In a right triangle, two of the altitudes are the legs of the triangle. B A K E A A D F F AB, AD, AF altitudes of right F I B D In an obtuse triangle, two of the altitudes are outside of the triangle. BI , DK , AF altitudes of obtuse 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. R In the isosceles ∆POQ, PR is the perpendicular bisector. Q