Triangle Fundamentals Lesson 3-1 Triangle Fundamentals

For example, we can call the following triangle: Classifying Triangles We name a triangle using its vertices. For example, we can call the following triangle: ∆ABC ∆ACB ∆BAC ∆BCA ∆CAB ∆CBA

Opposite Sides and Angles We say that is opposite .

Triangles can be classified by: Their sides Scalene Isosceles Equilateral Their angles Acute Right Obtuse Equiangular

A triangle in which all 3 sides are different lengths Scalene Triangle A triangle in which all 3 sides are different lengths

Isosceles Triangle A triangle in which at least 2 sides are equal

A triangle in which all 3 sides are equal Equilateral Triangle A triangle in which all 3 sides are equal

Acute Triangle A triangle in which all 3 angles are less than 90˚

Right Triangle A triangle in which exactly one angle is 90˚

Obtuse Triangle A triangle in which exactly one angle is greater than 90˚and less than 180˚

A triangle in which all 3 angles are the same measure Equiangular Triangle A triangle in which all 3 angles are the same measure

Classification of Triangles Flow Charts Venn Diagrams with Flow Charts and Venn Diagrams

Classification by Sides polygons Polygon triangles Triangle scalene isosceles Scalene Isosceles equilateral Equilateral

Classification by Angles polygons Polygon triangles Triangle right acute equiangular Right Obtuse Acute obtuse Equiangular

Triangle Theorems

Triangle Sum Theorem The sum of the interior angles in a triangle is 180˚.

Third Angle Corollary If two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent.

Corollary Each angle in an equiangular triangle is 60˚.

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

Corollary Acute angles in a right triangle are complementary.

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.

Special Segments of Triangles

Introduction There are three segments associated with triangles: Medians, Altitudes and Perpendicular Bisectors

Medians Definition: a segment from a vertex to the midpoint of the opposite side The median is in red here.

Medians How many medians does every triangle have? Since there are three vertices, there are three medians. Look at this example where D, E and F are the midpoints:

Altitudes Definition: the perpendicular segment from a vertex to the line that contains the opposite side Every triangle has three altitudes. See the altitudes:

Altitudes in a right triangle Two of the altitudes of a right triangle are also legs of the triangle. See this example: All of the altitudes are colored black.

Altitudes in an obtuse triangle Two of the altitudes here must be outside of the triangle. See this example: The original triangle is solid blue and the altitudes are black.

Perpendicular Bisector 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! Every triangle has three perpendicular bisectors.

Example 1: Draw the perpendicular bisector of in this scalene triangle. Notice how the bisector does not originate from a vertex of the triangle.

Example 2: See this example in a right triangle. Draw the perpendicular bisector of in this triangle.

Special Case The perpendicular bisector can pass through a vertex in an isosceles triangle (and therefore in an equilateral triangle). In the isosceles triangle, when the bisector is drawn to the base, it will pass through the vertex angle point.

Isosceles Triangle In , if is the base, draw the perpendicular bisector to . P is the vertex of the triangle. The perpendicular bisector will pass through P when drawn to the base.