Aptitude - Geometry
A point is an exact location
The straight path between two points A and B is called a line segment AB. A line segment has two end points.
On extending a line segment AB indefinitely in one direction we get the ray AB. Ray AB has one end point, namely A.
A line segment AB extended indefinitely in both directions is called line AB.
A line contains infinitely many points.
Through a given points , infinitaly many lines can be drawn.
One and only one line can be drawn to pass through two given points A and B.
Two line meet in a point.
Two planes meet in a line.
In the given figure, the points A,B,C are collinear.
Three or more lines intersecting at the same points are called concurrent lines.
Two rays OA and OB having a common end points O form angle AOB, written as ∠AOB
Measure of an Angle
The amount of turning from OA to OB is called the measure of ∠AOB written as m(∠AOB).
An angle of 360°
If a ray OA starting from its original position OA , rotates about O in anticlockwise direction and after a complete rotation comes back to its original position , then we say that it has rotated through 360. This complete rotation is divided into 360° equal parts. Then, each part is called 1 degree , written as 1°
1° = 60 minutes, written as 60'
1 minute = 60 seconds, written as 60"
Types of Angle
Right angle - An angle whose measure is 90° is called a right angle.
Acute angle - An angle whose measure is less than 90° is called an acute angle.
Obtuse angle - An angle whose measure is more than 90° but less than 180°, is called an obtues angle.
Straight angle - An angle whose measure is 180° is called a Straight angle.
Reflex angle - An angle whose measure is more than 180° but less than 360°, is called a Reflex angle.
Complete angle - An angle whose measure is 360°, is called a complete angle.
Equal angle - Two angles are said to be equal , if they have the same measure.
Complementary angleTwo angles are said to be complementary if the sum of their measures is 90. For example, angles measuring 65° and 25° are complementary angle.
Supplementary angle - Two angle are said to be supplementary if the sum of their measures is 180°. For example, angles measures 70° and 110° are supplementary.
Adjacent angle - Two angles are called adjacent angle if they have the same vertex and a common arm such that non-common arms are on either side of the comman arm. In the given figure , ∠AOC and ∠BOC are adjacent angle.
If a ray stands on a line , than the sum of two adjacent angle so formed is 180° In the given figure , ray CP stands on line AB.
∴ ∠ACD + ∠BCD = 180°.
The sum of all angle formed on the same side of a line at a given point on the line is 180°. In the given figure four angle are formed on the same side of AOB.
∴ ∠AOE + ∠EOD + ∠DOC + ∠COD = 180°.
The sum of all angle around a point is 360° In the given figure five angle are formed around a point O.
∴∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA=360°.
Vertically Opposite Angles
If two lines A Band CD intersect at a point O, then AOC , BOD and BOC , AOD are two pair of vertically opposites angle Vertically opposite angle are always equal.
∴ ∠AOC = ∠BOD and ∠AOD = ∠BOC
If two lines lie in the same plane and do not intersect when produced on either side then such lines are said to be paralleled and we write , L||m.
Traversal line cutting parallel lines
Let two parallel lines AB and CD be cut by a transversal EF. Then
Corresponding angle are equal
(∠1 = ∠5), (∠4= ∠8 ), (∠2 = ∠6) , (∠3 = ∠7)
Alternate interior angles are equal.
(∠3 =∠5 ) and (∠4 =∠6 )
Consective interior angles are supplementary
∠4+∠5 = 180° and ∠3 +∠6 = 180°.
A figure bounded by three straight lines is called a triangle. In the given figure , we have ∆ABC; ∆ABC having three vertices A,B,C. In has three angles, namely ∠A,∠B and ∠C. It has three sides , namely AB, AC and BC.
Types of Triangle
A triangle having all sides equal is called an equilateral triangle.
A triangle having two sides equal, is called an isosceles triangle.
A triangle having all sides of different lengths,is called a scalene triangle.
A triangle one of whose angles measures 90°,is called a right triangle.
A triangle one of whose angle lies between 90° and 180° is called an obtuse triangle.
A triangle each of whose angle is acute, is called an acute triangle.
The sum of all sides of a triangle is called the perimeter of the triangle.
The sum of two sides of a triangle is greater than the third side.
In a right angled ABC in which ∠B = 90°, we have AC2 =AB2+BC2. This is called Pythagoras Theorem.
A figure bounded by four straight line is called a quadrilateral. The sum of all angles of a quadrilateral is 360°.
Rectangle - A quadrilateral is called a rectangle, if its opposite side are equal and each of its angle is 90°. In given fig. ABCD is a rectangle.
Square - A quadrilateral is called a square, if all of its sides are equal and each of its angles measures 90°. In given fig. ABCD is square in which AB = BC = CD = DA.
Parallelogram - A quadrilateral is called a parallelogram, if its opposite sides are parallel. In given fig. ABCD is a parallelogram in which AB = DC & AD = BC.
Rhombus - A parallelogram having all sides equal is called a rhombus. In given fig. ABCD is a rhombus in which AB =BC =CD=DA, AB || DC and AD || BC.
A quadrilateral is a rectangle if opposite sides are equal and its diagonals are equal.
A quadrilateral is a Square if all sides are equal and the diagonal are equal.
A quadrilateral is a parallelogram, if opposite sides are equal.
A quadrilateral is a parallelogram but not a rectangle, if opposite sides are equal but the diagonals are not equal.
A quadrilateral is a rhombus but not a square if all their sides are equal and the diagonals are not equal.
Results on Quadrilateral
In a parallelogram, we have
Opposite sides are equal.
Opposite angles are equal.
Each diagonal bisects the parallelogram.
Diagonals of a parallelogram bisect each other.
Diagonals of a rectangle are equal.
Diagonals of a rhombus bisect each other at right angles.
Results on Circle
The perpendicular from the center to a chord bisects the chord.
There is one and only one circle passing through three non collinear points.
Angle in a semi circle is a right angle.
Opposite angles of a cyclic quadrilateral are supplementary.
Angle in the same segment of a circle is equal.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Two tangent to a circle from a point outside it are equal.
If PT is a tangent to a circle and PAB is a secant, Then PA x PB= PT2