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Triangle Fundamentals Intro to G.10 Modified by Lisa Palen 1

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Triangle Definition: A triangle is a three-sided polygon. What’s a polygon? 2

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These figures are not polygonsThese figures are 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. Polygons 3

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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. 4

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Triangles can be classified by: Their sides Scalene Isosceles Equilateral Their angles Acute Right Obtuse Equiangular 5

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Classifying Triangles by Sides Equilateral: Scalene: A triangle in which no sides are congruent. Isosceles: AB = 3.02 cm AC = 3.15 cm BC =3.55 cm A B C AB = 3.47 cm AC = 3.47 cm BC =5.16 cm B C A HI =3.70 cm G H I GH = 3.70 cm GI = 3.70 cm A triangle in which at least 2 sides are congruent. A triangle in which all 3 sides are congruent. 6

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Classifying Triangles by Angles Obtuse: Right: A triangle in which one angle is.... A triangle in which one angle is... 108 44 28 B C A obtuse. right. 7

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Classifying Triangles by Angles Acute: Equiangular: A triangle in which all three angles are.... A triangle in which all three angles are... acute. congruent. 57 47 76 G HI 8

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Classification of Triangles with Flow Charts and Venn Diagrams 9

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polygons Classification by Sides triangles Scalene Equilateral Isosceles Triangle Polygon scalene isosceles equilateral 10

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polygons Classification by Angles triangles Right Equiangular Acute Triangle Polygon right acute equiangular Obtuse obtuse 11

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Naming Triangles For example, we can call this triangle: We name a triangle using its vertices. ∆ABC ∆BAC ∆CAB∆CBA ∆BCA ∆ACB Review: What is ABC? 12

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Parts of Triangles For example, ∆ ABC has Sides: Angles: Every triangle has three sides and three angles. ACB ABC CAB 13

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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 14

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Interior Angle of a Triangle For example, ∆ ABC has interior angles: ABC, BAC, BCA An interior angle of a triangle (or any polygon) is an angle inside the triangle (or polygon), formed by two adjacent sides. 15

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Interior Angles Exterior Angle For example, ∆ABC has exterior angle ACD, because ACD forms a linear pair with ACB. 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. A B C D Exterior Angle 16

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Interior and Exterior Angles For example, ∆ ABC has exterior angle: ÐAÐACD and remote interior angles A and B 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. A B C D Exterior Angle Remote Interior Angles 17

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Triangle Theorems 18

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Triangle Sum Theorem The sum of the measures of the interior angles in a triangle is 180˚. m
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Third Angle Corollary If two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent. 20

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Third Angle Corollary Proof The diagramGiven: statementsreasons Prove: C F 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 4. Subtraction Property of Equality 5.Property: Substitution 6.Property: Substitution 7.Definition: congruence QED 21

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Corollary Each angle in an equiangular triangle measures 60˚. 60 22

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Corollary There can be at most one right or obtuse angle in a triangle. Example Triangles??? 23

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Corollary Acute angles in a right triangle are complementary. Example 24

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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. Exterior Angle Remote Interior Angles A B C D Example: Find the m A. 3x - 22 = x + 80 3x – x = 80 + 22 2x = 102 x = 51 m A = x = 51° 25

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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) 26

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Special Segments of Triangles 27

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Introduction There are four segments associated with triangles: Medians Altitudes Perpendicular Bisectors Angle Bisectors 28

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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. In the figure C, E and F are the midpoints of the sides of the triangle. B A D E C F 29

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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 A D F In an obtuse triangle, two of the altitudes are outside of the triangle. B A D F I K 30

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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! Example: C D In the scalene ∆CDE, is the perpendicular bisector. In the right ∆MLN, is the perpendicular bisector. In the isosceles ∆POQ, is the perpendicular bisector. E A B M L N AB R O Q P 31

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