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What is Geometry ?

The word geometry : Geo means earth and metry means to measure.
GEOMETRY means to study the position, shape, size and other properties of different figures.
Let us study about geometric plane figures and these have perfect shape and can be defined.

Eg: a rectangle has opposite sides equal, a unique definition
A square has all sides equal a unique definition etc like this if we can define a figure it is geometric figure eg: circle, triangle, quadrilateral all are geometric figures. Apart from the concepts of point, line, ray, line segment, surface, plane we go for following

1. Parallel Lines

Two straight lines are said to be parallel if they lie on the same plane and do not meet even when they are extended on either side. Distance between them is same at all points.

Parallel rays with equal distance

Eg: Two edges of a book are parallel
Two opposite sides of a rectangle/square are parallel

1.1 Concept of Transversal Lines

When a line cuts two or more parallel lines (or non parallel) at two different points it is called a transversal. But for a transversal the following pair of angles is equal

a) Corresponding angles are equal
b) Alternate angles are equal

c) Sum of interior angles is 180
d) Sum of exterior angles is 180

2. Perpendicular Lines

The two lines are said to be perpendicular to each other, if they contain angle of 90` between them.

Eg: All four corners of our room are 90 degrees

3. Congruence

Two figures are said to be congruent if they have same shape and same size

Eg: Two line segments of same length are congruent

Eg: two circles of same radius are congruent

3.1 Congruency in Triangles

SSS Congruence: Two triangles are said to be congruent if three sides are equal to corresponding three sides of other triangle.

SAS Congruence: If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, each to each then the triangles are SAS congruent.

ASA Congruence: If two angles and the included side of one triangle are equal to the two angles and the included side of the other triangle, then the triangles are ASA congruent.

RHS Congruence: If the hypotenuse and one side of a right angled triangle are equal to the hypotenuse and the one side of another right angled triangle, then the two triangles are RHS congruent.

Note : For congruency at least one pair of corresponding sides must be equal.
AAA is not test of congruency.
Example 1 state by what test the following triangles are congruent.

Here two angles are equal to the two angles of the other triangle and the included sides are equal. Therefore two triangles are congruent by ASA

Example 2

Here angle B = angle Q
BC = QR
Angle A = 180 – (20 + 50) = 110
Angle R = 180 – (20 +110) =50
Angle A = Angle R
Therefore two angles and included side of two triangles are equal
Δ ABC is congruent to Δ PQR (ASA congruent property)

TRY FOR YOURSELF

1) In the given figure , prove that :
Δ ABC = Δ ACD

2) In the given figure , prove that : BD = BC

3) In the given figure, prove that:

A ) Δ ACB is congruent to ΔECD
B ) AB = ED

4 ) Prove that :
a) Δ ABC is congruent to Δ ADC
b) Angle B = angle D

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Scarlett

About Scarlett

Bachelor in Mathematics, MSc in physics from the university of Calgary, Canada. Scarlett has been teaching mathematics to high school students from last 5 years.