Basic Geometrical Ideas

Lines

In our childhood, our first experience with a pencil would surely have been associated with drawing random lines. But, do you know lines are the most vital element of ancient geometry. Moreover, this geometric figure has led to the development of several modern day theories which we are studying at present. Let us try to relate to this concept and form an understanding of the significance of line in mathematics.

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Definition of Line

In a precise manner, a line doesn’t hold a beginning or end point. You can imagine it continuing infinitely in both directions. We can demonstrate it by little arrows trailing at both ends.

Line Segment

When two points are linked with a straight line, this is when we get a line segment. This below-mentioned line segment is AB.

Ray

A ray initiates from a point and lasts off to infinity. This can be represented by drawing an arrow symbol at one end of the ray. Sunrays can be a perfect example that initiates from the sun and travels indefinitely.

Acute Angle

From the figure drawn below, the angle falling between 0° and 90° is termed as an acute angle.

Obtuse Angle

An angle falling between 90° and 180° is an obtuse angle. From the figure, ∠B is an obtuse angle.

Right Angle

An angle that is 90° is called as a Right angle. In the figure, ∠C represents a right angle.

Supplementary Angles

Based on the figure, ∠AOC + ∠COB = ∠AOB = 180° If the addition of two angles is 180°; in this case, the angles are termed as supplementary angles.

Further, it should be noted that two right angles would always supplement each other. Also, the pair of adjacent angles which when added form a straight angle is termed as a linear pair.

Complementary Angles

Based on the figure, ∠COA + ∠AOB = 90°.

Hence, if the sum of two angles is 90°; in this case, the two angles are known as complementary angles.

Adjacent Angles

The angles which hold a common arm, as well as a common vertex, are termed as adjacent angles. Therefore, referring the above figure ∠BOA and ∠AOC are known as adjacent angles. OA is the common arm, with common vertex ‘O’.

Vertically Opposite Angles

Whenever two lines intersect, the formation of angles is opposite to each other specifically at the point of intersection (vertex). These are termed as vertically opposite angles.

From the above figure above, x and y are seen as the intersecting lines. ∠A and ∠C form one pair of vertically opposite angles, whereas, ∠B and ∠D is the other pair of vertically opposite angles.

Perpendicular Lines

Whenever there is a right angle in the middle of two lines, then the lines are known to be perpendicular to each other.

From the figure, the lines OA and OB are termed as perpendicular to each other.

Parallel Lines

Referring to the figure, A and B are the two parallel lines, which are intersected by a line p. Here, the line p is known as a transversal, which meets two or more lines at distinct points.

Transversal Intersecting Two Parallel Lines

Under this condition, one must remember that:

  • The corresponding angles tend to be equal.
  • Vertically opposite angles become equal.
  • Alternate exterior angles are equal.
  • Alternate interior angles are equal.
  • Pair of interior angles falling on the same side of the transversal is supplementary.

Question For You

Q. Suppose lines m and n tend to be parallel, then define the angles ∠5 and ∠7.

Solution: It is mentioned that, ∠2 = 125°
∠2 = ∠4 since vertically opposite angles. Hence, ∠4 = 125°
Now, observe that ∠4 is one of the interior angles falling on the same side of the transversal.
Thus, ∠4 + ∠5 = 180°
125 + ∠5 = 180 → ∠5 = 180 – 125 = 55°
∠5 = ∠7 because vertically opposite angles.
Therefore, ∠5 = ∠7 = 55°.

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