TRIANGLES

Nagesh Agrawal

PRESENTED BY….

…..CONTENTS…..• TRIANGLES

1. DEFINITION2. TYPES3. SECONDARY PART 4. CONGRUENCY5. SIMILARITY 6. AREA7. PYTHAGORAS THEOREM

TRIANGLES

A 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 TRIANGLESOn 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.

MEDIAN OF A TRIANGLE

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

MEDIAN

SECONDARY PARTS OF A TRIANGLE

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

1. SSS CRITERIA: If the three sides of one Triangle are equal to the three sides of another Triangle. Side-

Side-Side criteria of congruency.2. SAS CRITERIA : If two sides and the angle included

between them is equal to the corresponding two sides and the angle between them of another

triangle. Side-Angle-Side Criteria of Congruency.

CONGRUENCY OF A TRIANGLE

3. ASA CRITERIA : 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.4. RHS CRITERIA : If the hypotenuse, and a leg of one

right angled triangle is equal to corresponding hypotenuse and the leg of another right angled

triangle. Right Angle-Hypotenuse-Side Criteria of Congruency.

SIMILARITY CRITERIA OF A TRIANGLE

• AAA Similarity: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same

ratio (or proportion) and hence the two triangles are similar.

• SSS Similarity: If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides

of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

• SAS Similarity: If one angle of a triangle is equal to one angle of the other triangle and the sides including these

angles are proportional, then the two triangles are similar.

HERON’S FORMULAHeron’s Formula can be used in finding area of all types of Triangles. The Formula is :-

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

Pythagoras TheoremPythagoras' theorem, is a relation in Euclidean

geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the

side opposite the right angle) is equal to the sum of the squares of the other two sides.

Hypotenuse2 = Perpendicular2 + Base2

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