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Triangles

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Triangles

1. 1. TRIANGLES
2. 2. CONTENTS• TRIANGLES 1. DEFINITION 2. TYPES 3. PROPERTIES 4. SECONDARY PART 5. CONGRUENCY 6. AREA
3. 3. TRIANGLESA triangle is a 3-sided polygon. Every triangle has threesides, three vertices and three angles. On the basis of sidesof a triangle, triangles are of three types, An EquilateralTriangle, An Isosceles Triangle and A Scalene Triangle. Alltriangles are convex and bicentric. That portion of the planeenclosed by the triangle is called the triangle interior, whilethe remainder is the exterior.The study of triangles is sometimes known as triangle geometry and is a rich area of geometry filled with beautiful results and unexpected connections.
4. 4. TYPES OFTRIANGLES
5. 5. TYPES OF TRIANGLESOn Basis of Length of Sides, there are 3 types of Triangles• Equilateral Triangle• Isosceles Triangle• Scalene TriangleOn Basis of Angles, there are 3 types of triangles• Acute Angled Triangle• Obtuse Angled Triangle• Right Angled Triangle
6. 6. EQUILATERAL TRIANGLETriangles having all sides equal are called EquilateralTriangle. ISOSCELES TRIANGLE Triangles having 2 sides equal are called Isosceles Triangle.
7. 7. SCALENE TRIANGLETriangles having no sides equal are called ScaleneTriangle.
8. 8. ACUTE ANGLED TRIANGLETriangles whose all angles are acute angle arecalled Acute Angled Triangle. OBTUSE ANGLED TRIANGLE Triangles whose 1 angle is obtuse angle are called Obtuse Angled Triangle. RIGHT ANGLED TRIANGLETriangles whose 1 angle is right angle arecalled Right Angled Triangle.
9. 9. PROPERTIES OF A TRIANGLE
10. 10. PROPERTIES OF A TRIANGLETriangles are assumed to be two-dimensional planefigures, unless the context provides otherwise. In rigoroustreatments, a triangle is therefore called a 2-simplex.Elementary facts about triangles were presented by Euclidin books 1–4 of his Elements, around 300 BC.The measures of the interior angles of the triangle alwaysadd up to 180 degrees.
11. 11. PROPERTIES OF A TRIANGLEThe measures of the interior angles of a trianglein Euclidean space always add up to 180 degrees.This allows determination of the measure of thethird angle of any triangle given the measure oftwo angles. An exterior angle of a triangle is anangle that is a linear pair to an interior angle. Themeasure of an exterior angle of a triangle is equalto the sum of the measures of the two interiorangles that are not adjacent to it; this is theExterior Angle Theorem. The sum of themeasures of the three exterior angles (one foreach vertex) of any triangle is 360 degrees.
12. 12. ANGLE SUM PROPERTYAngle sum Property of a Triangle is that the sum ofall interior angles of a Triangle is equal to 180˚. EXTERIOR ANGLE PROPERTYExterior angle Property of a Triangle is that Anexterior angle of the Triangle is equal to sum of twoopposite interior angles of the Triangle.
13. 13. PYTHAGORAS THEOREMPythagoras Theorem is a theorem given byPythagoras. The theorem is that In a Right AngledTriangle the square of the hypotenuse is equal to thesum of squares of the rest of the two sides. HYPOTENUSE
14. 14. SECONDARY PARTS OF A TRIANGLE
15. 15. MEDIAN OF A TRIANGLEThe Line Segment joining the midpoint of the base ofthe Triangle is called Median of the Triangle.ORA Line Segment which connects a vertex of a Triangleto the midpoint of the opposite side is called Medianof the Triangle. MEDIAN
16. 16. ALTITUDE OF A TRIANGLEThe Line Segment drawn from a Vertex of a Triangle perpendicular to its opposite side is called an Altitude or Height of a Triangle. ALTITUDE
17. 17. PERPENDICULAR BISECTORA line that passes through midpoint of thetriangle or the line which bisects the thirdside of the triangle and is perpendicular to it iscalled the Perpendicular Bisector of thatTriangle. PERPENDICULAR BISECTOR
18. 18. ANGLE BISECTORA line segment that bisects an angle of atriangle is called Angle Bisector of the triangle. ANGLE BISECTOR
19. 19. CONGRUENCY OF A TRIANGLE
20. 20. SSS CRITERIA OF CONGRUENCYIf the three sides of one Triangle are equal tothe three sides of another Triangle. Then thetriangles are congruent by the SSS criteria.SSS criteria is called Side-Side-Side criteria ofcongruency.
21. 21. SAS CRITERIA OF CONGRUENCYIf two sides and the angle included betweenthem is equal to the corresponding two sidesand the angle between them of anothertriangle. Then the both triangles arecongruent by SAS criteria i.e. Side-Angle-SideCriteria of Congruency.
22. 22. ASA CRITERIA OF CONGRUENCYIf two angles and a side of a Triangle is equalto the corresponding two angles and a side ofthe another triangle then the triangles arecongruent by the ASA Criteria i.e. Angle-Side-Angle Criteria of Congruency.
23. 23. RHS CRITERIA OF CONGRUENCYIf the hypotenuse, and a leg of one rightangled triangle is equal to correspondinghypotenuse and the leg of another rightangled triangle then the both triangles arecongruent by the RHS criteria i.e. Right Angle-Hypotenuse-Side Criteria of Congruency.
24. 24. AREA OF ATRIANGLE
25. 25. HERON’S FORMULAHeron’s Formula can be used in finding area ofall types of Triangles. The Formula is ::->AREA =S = Semi-Perimetera,b,c are sides of the Triangle
26. 26. FORMULA FOR ISOSCELES TRIANGLEArea of an Isosceles Triangle =b = basea = length of equal sides
27. 27. FORMULA FOR RIGHT ANGLED TRIANGLE½ x base x height
28. 28. PYTHAGORAS EUCLID PASCALMATHEMATICIANS RELATED TO TRIANGLES
29. 29. THANKS