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. .
When two straight lines intersect, vertically
opposite angles are equal. In the figure given alongside, ∠AOB = ∠COD and ∠BOC = ∠AOD. .
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. .
of interior angles on the same side of the cutting line is equal to 180°. .
i.e. ∠QMN + ∠MNS = 180° and .
∠PMN + ∠MNR = 180°
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.