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Basic Theory - 3 on Plane Geometry for Banking n SSC

Posted on - 08-02-2017

QA Math Plane Geometry

Bank PO

PLANE GEOMETRY

The problems relating to geometry cover mostly triangles and circles. Even though polygons also are covered, the emphasis on polygons is not as much as on triangles and circles. In this chapter, we will look at some properties as well as theorems and riders on parallel lines, angles, triangles (including congruency and similarity of triangles), circles and polygons. .

Angles and Lines

An angle of 90° is a right angle; an angle less than 90° is an acute angle; an angle between 90° and 180° is an obtuse angle; and angle between Mr and 360° is a reflex angle. .

The sum of all angles on one side of a straight line AB at a point O by any number of lines joining the line AB at O is 180°. When any number of straight lines join at a point, the sum of all the angles around that point is 360°. .

Two angles whose sum is 90° are said to be complementary to each other and two angles whose sum is 180° are said to be supplementary angles. .

Fig. 1.

When two straight lines intersect, vertically opposite angles are equal. In the figure given alongside, ∠AOB = ∠COD and ∠BOC = ∠AOD. .

Parallel Lines

Fig. 2.

When a straight line XY cuts two parallel lines PQ and RS [as shown in the figure above], the following are the relationships between various angles that are formed. [M and N are the points of intersection of XY with PQ and RS respectively]. .

(a) Alternate angles are equal. .

i.e. ∠PMN = ∠MNS and ∠QMN = ∠MNR. .

(b) Corresponding angles are equal. .

i.e. ∠XMQ = ∠MNS; .

∠QMN = ∠SNY; ∠XMP = ∠MNR; and

∠PMN = ∠RNY. .

(c) Sum of interior angles on the same side of the cutting line is equal to 180°. .

i.e. ∠QMN + ∠MNS = 180° and .

∠PMN + ∠MNR = 180°

(d) Sum of exterior angles on the same side of the transversal is equal to 180°. .

i.e. ∠XMQ + ∠SNY = 180°; and .

∠XMP + ∠RNY = 180°. .

If three or more parallel lines make equal intercepts on one transversal, they make equal intercepts on any other transversal as well. The most general form of this is “if three or more parallel lines make intercepts on a transversal in a certain pro-portion, then they make intercepts in the same proportion on any other transversal as well”. .

If a line parallel to one side of a triangle intersects the other two sides in distinct points, theft it cuts off from these line segments proportional to these sides. The converse of this also is true. .

In particular, the line joining the midpoints of two sides of a triangle is parallel to and half of time third side.

 
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