Basics — Concept, Formulas & Examples

Fundamentals of Geometry

Point: It is an exact location. It is a fine dot which has neither length nor breadth nor thickness but has position i.e. it has no magnitude.

Line: A line is a set of all the points in one dimension. It can be extended in both the directions; hence, its length is infinite. A line does not have either width or thickness.

Intersecting lines: Two lines having a common point are called intersecting lines. The common point is known as the point of intersection.

Line Segment: The straight path joining two points A and B is called a line segment AB. It has end points and a definite length.

Ray: A line segment which can be extended in only one direction is called a ray.

Concurrent lines: If two or more lines intersect at the same point, then they are known as concurrent lines.

Plane: A plane is set of all the points in two dimensions. It does not have any thickness but is indefinitely extended in all directions.

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Angles: when two straight lines meet at a point they form an angle.

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In the figure above, the angle is represented as ∠AOB. OA and OB are the arms of ∠AOB. Point O is the vertex of ∠AOB. The amount of turning from one arm (OA) to other (OB) is called the measure of the angle (AOB).

Right angle: An angle whose measure is 90° is called a right angle.

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Acute angle: An angle whose measure is less than one right angle (i.e. less than 90°), is called an acute angle.

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Obtuse angle: An angle whose measure is more than one right angle and less than two right angles (i.e. less than 180° and more than 90°) is called an obtuse angle.

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Straight angle: An angle that measure exactly 180° is called a straight angle. The angle shown in the figure below is a straight angle.

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Reflex angles: An angle whose measure is more than 180° and less than 360° is called a reflex angle.

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Complementary angles: If the sum of the two angles is one right angle (i.e. 90°), they are called complementary angles. Therefore, the complement of an angle θ is equal to 90° - θ.

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Supplementary angles: Two angles are said to be supplementary, if the sum of their measures is 180°. Therefore, supplement of an angle θ is 180° - θ.

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Bisector of an angle: If a ray or a straight line passing through the vertex of that angle, divides the angle into two angles of equal measurement, then that line is known as the bisector of that angle.

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A point on an angle bisector is equidistant from both the arms.

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In the figure above, Q and R are the feet of perpendiculars drawn from P to OB and OA. It follows that PQ = PR.

Vertically opposite angles: When two straight lines intersect each other at a point, the pairs of opposite angles so formed are called vertically opposite angles.

In the above figure, ∠1 and ∠3, ∠2 and ∠4 are vertically oppostie angles.

Vertically opposite angles are always equal.

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Adjacent angle: When two angles share a common side and a common vertex, we get two adjacent angles. For two angles to be adjacent, no angles should be inside the other.

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If the sum of two adjacent angles is 180° then these angles form a linear pair.

Angles making a linear pair are supplementary to each other.

Perpendicular lines: Two lines intersecting each other at 90° are said to be perpendicular to each other.

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Parallel lines: Two lines are parallel if they are coplanar and they do not intersect each other even if they are extended on either side.

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Two lines in the same plane that are perpendicular to a given line are parallel to each other.

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Line segment bisector: A line segment bisector which makes an angle of 90° with the given segment is known as the perpendicular bisector for the given segment.

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Any point of the perpendicular bisector is at an equal distance from both the ends of the given line segment.

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Distance of a point from a line means the length of the perpendicular drawn from the point to the line. It is the shortest distance of the point from the line.

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Distance between two parallel lines is the perpendicular distance between them.

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Distance between two coplanar but non-parallel lines is always zero, because these lines intersect each other at some point.

Transversal: A transversal is a line that intersects (or cuts) two or more coplanar lines at distinct points.

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In the above, figure, a transversal t is intersecting two parallel lines, l and m, at A and B, respectively.

Angles formed by a transversal of two parallel lines:

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In the above figure, L₁ and L₂ are two parallel lines intersected by a transversal. The following properties of the angles can be observed: 

• Vertically opposite angles are equal: a = d, b = c, e = h and f = g
• Alternate interior angles are equal: c = f and d = e
• Alternate exterior angles are equal: a = h and b = g
• Corresponding angles are equal: a = e, b = f, c = g and d = h
• Interior angles on the same side of the transversal are supplementary: c + e = 180° and d + 180°
• Exterior angles on the same side of the transversal are supplementary: a + g = 180° and b + h = 180°

 

Basic Proportionality Theorem

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For a set of three or more parallel lines (L₁, L₂ and L₃ for example), and two transversals (T₁ and T₂ for example), the ratio of the lengths of the intercepts of any transversal is equal 

i.e., $\frac{\mathrm{AB}}{\mathrm{BC}}$ = $\frac{\mathrm{PQ}}{\mathrm{QR}}$