TRIANGLES

SHYAM

ADAMYAPRESENTED BY

CONTENTS• TRIANGLES

1. DEFINITION2. TYPES3. PROPERTIES 4. SECONDARY PART 5. CONGRUENCY6. AREA

TRIANGLESA triangle is a 3-sided polygon. Every triangle has three sides, three vertices and three angles. On the basis of sides of a triangle, triangles are of three types, An Equilateral Triangle, An Isosceles Triangle and A Scalene Triangle. All triangles are convex and bicentric. That portion of the plane enclosed by the triangle is called the triangle interior, while the 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.

TYPES OF

TRIANGLES

TYPES OF TRIANGLES

On Basis of Length of Sides, there are 3 types of Triangles• Equilateral Triangle• Isosceles Triangle• Scalene Triangle

On Basis of Angles, there are 3 types of triangles • Acute Angled Triangle• Obtuse Angled Triangle• Right Angled Triangle

EQUILATERAL TRIANGLETriangles having all sides equal are called Equilateral Triangle.

ISOSCELES TRIANGLETriangles having 2 sides equal are called Isosceles Triangle.

SCALENE TRIANGLETriangles having no sides equal are called Scalene Triangle.

Triangles whose all angles are acute angle are called Acute Angled Triangle.

ACUTE ANGLED TRIANGLE

RIGHT ANGLED TRIANGLE

OBTUSE ANGLED TRIANGLETriangles whose 1 angle is obtuse angle are called Obtuse Angled Triangle.

Triangles whose 1 angle is right angle are called Right Angled Triangle.

PROPERTIESOF A

TRIANGLE

PROPERTIES OF A TRIANGLE Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise. In rigorous treatments, a triangle is therefore called a 2-simplex. Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements, around 300 BC.The measures of the interior angles of the triangle always add up to 180 degrees.

PROPERTIES OF A TRIANGLE

The measures of the interior angles of a triangle in Euclidean space always add up to 180 degrees. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An exterior angle of a triangle is an angle that is a linear pair to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the Exterior Angle Theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees.

ANGLE SUM PROPERTY

Angle sum Property of a Triangle is that the sum of all interior angles of a Triangle is equal to 180˚.

EXTERIOR ANGLE PROPERTY

Exterior angle Property of a Triangle is that An exterior angle of the Triangle is equal to sum of two opposite interior angles of the Triangle.

PYTHAGORAS THEOREM

Pythagoras Theorem is a theorem given by Pythagoras. The theorem is that In a Right Angled Triangle the square of the hypotenuse is equal to the sum of squares of the rest of the two sides.

HYPOTENUSE

SECONDARY PARTS OF A TRIANGLE

MEDIAN OF A TRIANGLE

The Line Segment joining the midpoint of the base of the Triangle is called Median of the Triangle.

OR

A Line Segment which connects a vertex of a Triangle to the midpoint of the opposite side is called Median of the Triangle.

MEDIAN

ALTITUDE OF A TRIANGLE

The Line Segment drawn from a Vertex of a Triangle perpendicular to its opposite side is called an Altitude or Height of a Triangle.

ALTITUDE

PERPENDICULAR BISECTOR

A line that passes through midpoint of the triangle or the line which bisects the third side of the triangle and is perpendicular to it is called the Perpendicular Bisector of that Triangle.

PERPENDICULAR BISECTOR

ANGLE BISECTOR

A line segment that bisects an angle of a triangle is called Angle Bisector of the triangle.

ANGLE BISECTOR

CONGRUENCY OF A

TRIANGLE

SSS CRITERIA OF CONGRUENCY

If the three sides of one Triangle are equal to the three sides of another Triangle. Then the triangles are congruent by the SSS criteria.SSS criteria is called Side-Side-Side criteria of congruency.

SAS CRITERIA OF CONGRUENCY

If two sides and the angle included between them is equal to the corresponding two sides and the angle between them of another triangle. Then the both triangles are congruent by SAS criteria i.e. Side-Angle-Side Criteria of Congruency.

ASA CRITERIA OF CONGRUENCY

If two angles and a side of a Triangle is equal to the corresponding two angles and a side of the another triangle then the triangles are congruent by the ASA Criteria i.e. Angle-Side-Angle Criteria of Congruency.

RHS CRITERIA OF CONGRUENCY

If the hypotenuse, and a leg of one right angled triangle is equal to corresponding hypotenuse and the leg of another right angled triangle then the both triangles are congruent by the RHS criteria i.e. Right Angle-Hypotenuse-Side Criteria of Congruency.

AREA OF A

TRIANGLE

HERON’S FORMULA

Heron’s Formula can be used in finding area of all types of Triangles. The Formula is ::->

AREA = S = Semi-Perimeter a,b,c are sides of the Triangle

FORMULA FOR ISOSCELES TRIANGLE

Area of an Isosceles Triangle =

b = basea = length of equal sides

FORMULA FOR RIGHT ANGLED TRIANGLE

½ x base x height

MATHEMATICIANS RELATED TO TRIANGLES

PYTHAGORAS EUCLID PASCAL

THANKS