Naming Triangles Triangles are named by using its vertices. For example, we can call the following triangle: A B C ∆ABC ∆ACB ∆BAC ∆BCA ∆CAB ∆CBA

Opposite Sides and Angles Side opposite to A : Side opposite to B : Side opposite to C : Opposite Angles: Angle opposite to : A Angle opposite to : B Angle opposite to : C

Classifying Triangles by Sides Scalene: A triangle in which all 3 sides are different lengths. BC = 5.16 cm B C A BC = 3.55 cm A B C AB = 3.47 cm AC = 3.47 cm AB = 3.02 cm AC = 3.15 cm Isosceles: A triangle in which at least 2 sides are equal. HI = 3.70 cm G H I Equilateral: A triangle in which all 3 sides are equal. GI = 3.70 cm GH = 3.70 cm

Classifying Triangles by Angles Acute: A triangle in which all 3 angles are less than 90˚. 57 ° 47 76 G H I Obtuse: 108 ° 44 28 B C A A triangle in which one and only one angle is greater than 90˚& less than 180˚

Classifying Triangles by Angles Right: A triangle in which one and only one angle is 90˚ Equiangular: A triangle in which all 3 angles are the same measure.

Classification by Sides with Flow Charts & Venn Diagrams polygons Polygon triangles Triangle scalene isosceles Scalene Isosceles equilateral Equilateral

Classification by Angles with Flow Charts & Venn Diagrams polygons Polygon triangles Triangle right acute equiangular Right Obtuse Acute obtuse Equiangular

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. There can be at most one right or obtuse angle in a triangle. Corollary 3:

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 A Exterior Angle D Example: Find the mA. B C 3x - 22 = x + 80 3x – x = 80 + 22 2x = 102 mA = x = 51°

Median - Special Segment of Triangle Definition: A segment from the vertex of the triangle to the midpoint of the opposite side. B A D E C F Since there are three vertices, there are three medians. In the figure C, E and F are the midpoints of the sides of the triangle.

Altitude - Special Segment of Triangle The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side. Definition: B A D F In a right triangle, two of the altitudes are the legs of the triangle. B A D F I K In an obtuse triangle, two of the altitudes are outside of the triangle.

Perpendicular Bisector – Special Segment of a triangle A line (or ray or segment) that is perpendicular to a segment at its midpoint. Definition: The perpendicular bisector does not have to start from a vertex! R O Q P Example: M L N C D A E A B B In the isosceles ∆POQ, is the perpendicular bisector. In the scalene ∆CDE, is the perpendicular bisector. In the right ∆MLN, is the perpendicular bisector.