Geometry Tutorial Part - 1

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Geometry Tutorial Part - 1

Author: Sureshbala

In GMAT, GRE, SAT, CAT and other similar Entrance Exams, the problems relating to Geometry cover mostly triangles, quadrilaterals and circles. Even though polygons with more than four sides are also covered, the emphasis on such polygons is not as much as it is 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), quadrilaterals, circles and polygons.

ANGELS AND LINES

An angle of $90^{\circ}$ is a right angle; an angle less than $90^{\circ}$ is acute angle; an angle between $90^{\circ}$ and $180^{\circ}$ is an obtuse angle; $180^{\circ}$ and $360^{\circ}$ angle between and is a reflex angle.

The sum of all angles made on one side of a straight line AB at a point O by any number of lines joining the line AB at O is $180^{\circ}$ . In Fig. 1.01 below, the sum of the angles u, v, x, y and z is equal to $180^{\circ}$ .

When any number of straight lines join at a point, the sum of all the angles around that point is $360^{\circ}$ . In Fig 1.02 below, the sum of the angles u, v, w, x, y, and z is equal $360^{\circ}$ .

Two angles whose sum is $90^{\circ}$ are said to be complementary angles and two angles whose sum $180^{\circ}$ is are said to be supplementary angles.

When two straight lines intersect, vertically opposite angles are equal. In Fig.1.03 given below, $\angle AOB$ and $\angle COD$ are vertically opposite angles and $\angle BOC$ and $\angle AOD$ are vertically opposite angles. So, we have $\angle AOB = \angle COD$ and $\angle BOC = \angle AOD$

Two lines which make an angle of $90^{\circ}$ with each other are said to be PERPENDICULAR to each other.

If a line l_1 passes through the mid-point of another line l_2 then the line l_1 is said to be the BISECTOR of the line l_2 i.e., the line l_2 is divided into two equal parts.

If a line l_1 is drawn at the vertex of an angle dividing the angle two equal parts, then the line l_1 is said to be the ANGULAR BISECTOR of the angle. Any point on the angular bisector of an angle is EQUIDISTANT from the two arms of the angle.

If a line l_1 is perpendicular to line l_2 as well as passes through the mid-point of line l_2 , then the line l_1 is said to be the PERPENDICULAR BISECTOR of the line l_2 .

Any point on the perpendicular bisector of a line is EQUIDISTANT from both ends of the line.

In Fig 1.04, line PQ is the perpendicular bisector of line AB. A point P on the perpendicular bisector of AB will be equidistant from A and B, i.e., PA = PB Similarity. For any point R on the perpendicular bisector PQ, RA = RB.

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