Concepts of Geometry

Triangle

Definition : A triangle is a polygon with three sides.

                    It also has three vertices and three angles.

Types of triangle:

1. Equilateral Triangle

    An equilateral triangle is a triangle in which all three sides are the same length.

2. Isosceles Triangle:

    An isosceles triangle is a triangle in which two sides are the same length and the third side is a different length.

3. Scalene Triangle:

    A scalene triangle is a triangle in which all three sides are different lengths.

4. Acute Triangle

    An acute triangle is a triangle in which all three of the angles are acute angles.

 

5. Obtuse Triangle

    An obtuse triangle is a triangle in which one of the angles is a obtuse angle.

6. Right Triangle

    A right triangle is a triangle in which one of the angles is a right angle.

• Pythagoras theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.

Properties

• The sum of the interior angles of a triangle is 180 degrees.
• The sum of the exterior angles of a triangle is 360 degrees.
•  The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior     angles of the triangle.
•  Each interior opposite angle always less than that of exterior angle.
•   Angles opposite to two equal sides of a triangle are equal.
•  Sides opposite to two equal angles of a triangle are equal.
•  In an isosceles triangle altitude bisect the base.
•  If two sides of a triangle are unequal, then the longer side has greater angle opposite it.
• The greater angle of a triangle has the longer side opposite to it.
• The sum of any two side of a triangle is greater than the third side.
•  In a right angle triangle, the hypotenuse is the longest side.
• The sum of the three altitudes of a triangle is less than the sum of the three sides of the triangle.
•  The point of concurrency of the bisector of the angles of a triangle is called incenter.
•  The point of concurrency of the perpendicular bisector of the sides of a triangle is called circumcenter.
•  The point of concurrency of the three altitudes of a triangle is called orthocenter.
•   Median is a line segment joining a vertex of a triangle to the midpoint of its opposite side.
•  The medians of triangle passes through the same point known as centroid of triangle and it divides each median in the ratio of 2:1.
•  Triangles on the same base and between the same parallels are equal in area.
•  A median divides a triangle into two triangles of equal area.
•  If equilateral triangles are drawn on the sides of a right angle triangle, then the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.
•  If a perpendicular drawn from the opposite vertex to the hypotenuse of right triangle, the triangles on the opposite sides of the perpendicular are similar to each other.
•  The line segment joining the midpoint of the hypotenuse of a right triangle to the vertex of the right angle is equal to half the hypotenuse.

Theorem about Similarity of triangles

Two triangles are said to be similar if

• If two corresponding internal angles of two triangles have the same measure, the triangles are similar.
• If two corresponding sides of two triangles are in proportion, and their included angles have the same measure, then the triangles are similar. (The included angle for any two sides of a polygon is the internal angle between those two sides.)
• If three corresponding sides of two triangles are in proportion, then the triangles are similar.

Properties of Similar Triangles

•   If the corresponding sides of two triangles are proportional, then the triangles are similar.
•  The ratio of area of similar triangles is equal to the ratio of the square of the corresponding sides/altitudes/angle bisectors and medians.
•  The ratio of perimeters of similar triangles is equal to the ratio of the square of the corresponding sides/altitudes/angle bisectors and medians.
• Ratio of corresponding altitudes of two similar triangles is equal to the ratio of their corresponding sides.    

Congruence Axiom

1. SAS (Side-Angle-Side) Congruence theorem:

The two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding sides and angle of other triangle.

2. ASA (Angle-Side- Angle) Congruence theorem:

The two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding angles and side of other triangle.

3. SSS (Side-Side-Side) Congruence theorem:

The two triangles are congruent if three sides of one triangle are equal to the corresponding three sides of other triangle.

4. AAA (Angle-Angle- Angle) Congruence theorem:

The two triangles are congruent if three angles of one triangle are equal to the corresponding three angles of other triangle.

5. RHS (Right-Hypotenuse-Side) Congruence theorem:

Two right triangles are congruent if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and the corresponding side.

Properties

• The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and  equal to half of it.
• If a line is drawn parallel to one side of a triangle the other two sides are divided in the same ratio.
• If two lines intersect, then vertically opposite angles are equal.
• If one of the four angles formed by two intersecting lines is a right angle, then each of the four angles is a right angle.
• If a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.
• If a transversal intersects two parallel lines, then each pair of alternate and corresponding angles are equal.
• If a transversal intersect two lines in such a way that a pair of consecutive interior angles are supplementary, then two lines are parallel.
• If two lines are perpendicular to the same line, then they are parallel to each other.
• If two parallel lines are interested by a transversal, then bisectors of any two corresponding angles and alternate angles are parallel.
• Two line segments are congruent if and only if their lengths are equal.
• Two line angles are congruent if and only if their measures are equal.

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Quadrilateral

• In a regular polygon, all sides and interior angles are equal.
• The sum of the interior angles of a quadrilateral is 360 degrees.
• The exterior angle of a regular polygon is 360 degree/n.
• The interior angle of a regular polygon is (n-2/n) x 180 degree.
• The perimeter of a quadrilateral is greater than the sum of diagonals.

1. Parallelogram

A quadrilateral with each pair of opposite sides parallel.

Properties:

ØThe opposite angles are equal.

Ø  The opposite angles are equal.

Ø  The diagonals bisect each other.

2. Rhombus

A quadrilateral with each pair of opposite sides parallel and all the four sides are equal.

Properties:

Ø  The opposite angles are equal.

Ø  The diagonals perpendicularly bisect each other.

3. Rectangle

A quadrilateral with each pair of opposite sides parallel and equal.

Properties:

Ø  All four angles are right angles.

Ø  The diagonals are of equal length.

Ø  The diagonals bisect each other.

4. Square

A quadrilateral with each pair of opposite sides parallel and all the four sides are equal.

Properties:

Ø  All four angles are right angles.

Ø  The diagonals are of equal length.

Ø  The diagonals perpendicularly bisect each other.

5. Trapezium

A quadrilateral with one pair of the opposite sides is parallel.

Property:

The diagonal of a trapezium cut each other in the same ratio.     
    

6. Kite

A quadrilateral with exactly two pair of equal consecutive sides.
Property:

Ø  The diagonal are perpendicular to one another.

Ø  One of the diagonal bisects the other.

Ø  In the figure  < B = < D  But < A not equal to < C

•    The quadrilateral formed by joining the mid points of   consecutive sides of a quadrilateral is a parallelogram.
•    A diagonal of a parallelogram divides it into two triangles of equal area. 
•    Diagonals of a parallelogram divide it into four triangles of equal area.   
•     Any straight line parallel to the parallel sides of a trapezium cuts the other two sides proportionally.