Triangle Fundamentals Intro to G.10 Triangle Fundamentals Modified by Lisa Palen

Triangle What’s a polygon? Definition: A triangle is a three-sided polygon. What’s a polygon?

Polygons Definition: A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints. These figures are not polygons These figures are polygons

Definition of a Polygon A polygon is a closed figure in a plane formed by a finite number of segments that intersect only at their endpoints.

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

Classifying Triangles by Sides Scalene: A triangle in which no sides are congruent. 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 congruent. HI = 3.70 cm G H I Equilateral: A triangle in which all 3 sides are congruent. GI = 3.70 cm GH = 3.70 cm

Classifying Triangles by Angles Obtuse: 108 ° 44 28 B C A A triangle in which one angle is.... obtuse. Right: A triangle in which one angle is... right.

Classifying Triangles by Angles Acute: 57 ° 47 76 G H I A triangle in which all three angles are.... acute. Equiangular: A triangle in which all three angles are... congruent.

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

Naming Triangles We name a triangle using its vertices. For example, we can call this triangle: ∆ABC ∆ACB Review: What is ABC? ∆BAC ∆BCA ∆CAB ∆CBA

Parts of Triangles Every triangle has three sides and three angles. For example, ∆ABC has Sides: Angles:  CAB  ABC  ACB

Opposite Sides and Angles Side opposite of BAC : Side opposite of ABC : Side opposite of ACB : Opposite Angles: Angle opposite of : BAC Angle opposite of : ABC Angle opposite of : ACB

Interior Angle of a Triangle An interior angle of a triangle (or any polygon) is an angle inside the triangle (or polygon), formed by two adjacent sides. For example, ∆ABC has interior angles:  ABC,  BAC,  BCA

Exterior Angle An exterior angle of a triangle (or any polygon) is an angle that forms a linear pair with an interior angle. They are the angles outside the polygon formed by extending a side of the triangle (or polygon) into a ray. Exterior Angle Interior Angles A For example, ∆ABC has exterior angle ACD, because ACD forms a linear pair with ACB. D B C

Interior and Exterior Angles The remote interior angles of a triangle (or any polygon) are the two interior angles that are “far away from” a given exterior angle. They are the angles that do not form a linear pair with a given exterior angle. For example, ∆ABC has exterior angle: ACD and remote interior angles A and B Exterior Angle Remote Interior Angles A D B C

Triangle Theorems

m<A + m<B + m<C = 180 Triangle Sum Theorem The sum of the measures of the interior angles in a triangle is 180˚. m<A + m<B + m<C = 180 IGO GeoGebra Applet

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

Third Angle Corollary Proof Given: The diagram Prove: C  F statements reasons 1. A  D, B  E 2. mA = mD, mB = mE 3. mA + mB + m C = 180º mD + mE + m F = 180º 4. m C = 180º – m A – mB m F = 180º – m D – mE 5. m C = 180º – m D – mE 6. mC = mF 7. C  F 1. Given 2. Definition: congruence 3. Triangle Sum Theorem Subtraction Property of Equality Property: Substitution Definition: congruence QED

Corollary Each angle in an equiangular triangle measures 60˚. 60 60

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

Corollary Acute angles in a right triangle are complementary. Example

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 x = 51 mA = x = 51°

Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. GeoGebra Applet (Theorem 1)

Special Segments of Triangles

Introduction There are four segments associated with triangles: Medians Altitudes Perpendicular Bisectors Angle Bisectors

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.