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

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Presentation on theme: "Triangle Fundamentals Intro to G.10 Modified by Lisa Palen 1."— Presentation transcript:

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

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

4 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

5 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

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

7 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

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

9 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

10 Classification of Triangles with Flow Charts and Venn Diagrams 9

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

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

13 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

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

15 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

16 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

17 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

18 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

19 Triangle Theorems 18

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

21 Third Angle Corollary If two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent. 20

22 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

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

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

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

26 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 x – x = x = 102 x = 51 m  A = x = 51° 25

27 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

28 Special Segments of Triangles 27

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

30 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

31 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

32 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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