Lesson Triangle 3 1 Fundamentals Modified by Lisa

Lesson Triangle 3 -1 Fundamentals Modified by Lisa Palen

Triangle 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. From Lesson 3 -4

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

Classifying Triangles by Sides cm 47 3. = B A cm =3. 02 47 3. AB = C C BC = 3. 55 cm m 5 c 3. 1 B A A A = AC cm Scalene: A triangle in which no sides are congruent. B C BC = 5. 16 cm Isosceles: A triangle in which at least 2 sides are congruent. cm GH = 3. 70 cm H Lesson 3 -1: Triangle Fundamentals G . 70 =3 A triangle in which all 3 sides are congruent. GI Equilateral: 6 HI = 3. 70 cm I

Classifying Triangles by Angles Obtuse: A 44° A triangle in which one angle is. . obtuse. B Right: 28° 108° C A triangle in which one angle is. . . right. Lesson 3 -1: Triangle Fundamentals 7

Classifying Triangles by Angles Acute: G 76° A triangle in which all three angles are. . acute. 57° 47° H Equiangular: I A triangle in which all three angles are. . . congruent. Lesson 3 -1: Triangle Fundamentals 8

Classification of Triangles with Flow Charts and Venn Diagrams

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

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

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

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

14 Opposite Sides and Angles Opposite Sides: 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 Lesson 3 -1: Triangle Fundamentals

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. Interior Angles For example, ∆ABC has exterior angle ACD, because ACD forms a linear pair with ACB. B Exterior Angle A D 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, ∆ABCRemote has Interior exterior angle: Angles Exterior Angle A ACD and remote interior angles A and B B D C

Triangle Theorems

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 Geo. Gebra 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 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 QED reasons 1. Given 2. Definition: congruence 3. Triangle Sum Theorem 4. Subtraction Property of Equality 5. 6. 7. Property: Substitution Definition: congruence

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 Example: Find the m A. 3 x - 22 = x + 80 3 x – x = 80 + 22 2 x = 102 x = 51 B A Exterior Angle D C 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. Geo. Gebra Applet (Theorem 1)

Special Segments of Triangles

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

29 Median - Special Segment of Triangle Definition: A segment from the vertex of the triangle to the midpoint of the opposite side. Since there are three vertices, there are three medians. B C F D E A In the figure C, E and F are the midpoints of the sides of the triangle. Lesson 3 -1: Triangle Fundamentals

30 Altitude - Special Segment of Triangle Definition: The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side. In a right triangle, two of the altitudes are the legs of the triangle. B F I A K A D D In an obtuse triangle, two of the altitudes are outside of the triangle. Lesson 3 -1: Triangle Fundamentals

31 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: E M A A C D B In the scalene ∆CDE, is the perpendicular bisector. L B O N Q R In the isosceles ∆POQ, is In the right ∆MLN, is the perpendicular bisector. Lesson 3 -1: Triangle Fundamentals