Quantitative Aptitude > Lines and angles

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Lines and Angles

  • A point has no size and is the simplest geometric figure. All geometric figures consist of points.
  • A line is understood to be a straight line that extends in both directions without ending.
  • A plane can be thought of as a floor or a tabletop, except that a plane extends in all directions without ending and has no thickness.
  • Given any two points on a line, a line segment is the part of the line that contains the two points and all the points between them. The two points are called endpoints. Line segments that have equal lengths are called congruent line segments. The point that divides a line segment into two congruent line segments is called the midpoint of the line segment.

    In the figure below, A, B, C, and D are points on line .


    Line segment AB consists of points A and B and all the points on the line between A and B. Sometimes the notation AB denotes line segment AB, and sometimes it denotes the length of line segment AB. The meaning of the notation can be determined from the context. According to the figure above, the lengths of line segments AB, BC, and CD are 8, 6, and 6, respectively. Hence, line segments BC and CD are congruent. Since C is halfway between B and D, point C is the midpoint of line segment BD.

    • When two lines intersect at a point, they form four angles, as indicated below. Each angle has a vertex at point P, which is the point of intersection of the two lines.

    In the figure, angles APC and BPD are called opposite angles, also known as vertical angles. Angles APD and CPB are also opposite angles. Opposite angles have equal measures, and angles that have equal measures are called congruent angles. Hence, opposite angles are congruent. The sum of the measures of the four angles is .

    Sometimes the angle symbol is used instead of the word "angle." For e.g. angle APC can be written as APC.

    • Two lines that intersect to form four congruent angles are called perpendicular lines. Each of the four angles has a measure of  . An angle with a measure of is called a right angle. The figure below shows two lines, .

    • An angle with a measure less than  is called an acute angle, and an angle with measure between  and  is called an obtuse angle.
    • Two lines in the same plane that do not intersect are called parallel lines. The figure below shows two lines,  and , that are parallel, denoted by . The two lines are intersected by a third line, , forming eight angles. Note that four of the angles have the measure , and the remaining four angles have the measure , where .


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