Angles

Objective

This topic gives an overview of;

  • Angles
  • Acute Angle
  • Right Angle
  • Obtuse Angle
  • Straight Angle
  • Reflex Angle
  • Complementary Angles
  • Supplementary Angles
  • Adjacent Angles
  • Linear Pair
  • Vertically Opposite Angles

Angles

A line segment has two end points. If we extend the two end points in either direction endlessly, we get a line. Thus, we can say that a line has no end points. On the other hand, a ray has one end point (namely its starting point). An angle is formed when lines or line segments meet. The corners are formed when two lines or line segments intersect at a point.

 

In the example (i) line segments AB and BC intersect at B to form angle ABC, and again line segments BC and AC intersect at C to form angle ACB and so on. Whereas, in the example (ii) lines PQ and RS intersect at O to form four angles POS, SOQ, QOR and ROP.

The angle is represented by the symbol “ ∠ “. The lines that form an angle are called the sides or the arms of the angle. The common end point is called the vertex of the angle.

Types of Angles

Acute Angle

Acute angle is an angle formed by two lines or line segments, where the measure of the angle is less than 90 degree. Note that the acute angle does not quite reach 90 degrees; they are always less than 90 degrees. These angles appear "sharp," like the blade on a knife. Below depicts an example of acute angle.

 

Right Angle

A right angle is an angle whose measure is exactly 90 degree. Right angles are denoted by a small square in its interior. These are like the edges of a wooden block. Below depicts an example of right angle.

Obtuse Angle

Obtuse is an angle formed by two lines or line segments, where the measure of the angle is greater than 90 degree and less than 180 degree. Thus, it is between 90 degrees and 180 degrees. These are "fat" angles that are very wide. Below depicts an example of obtuse Angle.

 

Straight Angle

A straight angle is one whose measure is exactly 180 degrees. A straight angle is made up of two opposite rays or line segments. Another important fact is that a straight angle forms a straight line. Below depicts an example of Straight angle.

Reflex Angle

A reflex angle is one whose measure is greater than 180 and less than 360 degrees. The reflex angle is the larger angle. Below example depicts an example of Reflex Angle.

 

Complementary Angles

When the sum of the measures of two angles is 90 degree, the angles are called complementary angles. Whenever two angles are complementary, each angle is said to be the complement of the other angle.

In this example the sum of the measures of the angles in above pairs comes out to be 90 degree.

Supplementary Angles

When the sum of the measures of two angles is 180 degree, the angles are called supplementary angles. Whenever two angles are supplementary, each angle is said to be the supplement of the other angle.

In this example the sum of the measures of the angles in above pairs comes out to be 180 degree.

Adjacent Angles

Two angles which share the same vertex (centre, usually represented by 0) and have a common side (line) are called Adjacent angles. Adjacent angles have a common vertex and a common arm but no common interior points.

Adjacent angles are such that:

(i) They have a common vertex;
(ii) They have a common arm; and
(iii) The non-common arms are on either side of the common arm.

Below depicts an example of Adjacent Angles.

Linear Pair

A linear pair is a pair of adjacent angles whose non-common sides are opposite rays. The angles must be adjacent. If two angles form a linear pair, they are supplementary. If two congruent angles form a linear pair, they are right angles.

Real time examples are vegetable chopping board and a pen stand. The chopping blade makes a linear pair of angles with the board. The pen makes a linear pair of angles with the stand. Below depicts an example of linear pair of adjacent angles.

Vertically Opposite Angles

Vertically opposite angles are the angles opposite to each other when two straight lines intersect. Their defining property is that, vertically opposite angles are equal in magnitude. Vertically opposite angles are opposite to each other and are equal. Vertical angles are not adjacent. They are located across from one another in the corners of the "X" formed by the two straight lines. Below depicts an example of Vertically Opposite Angles.

 

Summary

  • We recall that,
    • A line-segment has two end points.
    • A ray has only one end point (its vertex)
    • A line has no end points on either side.
  • An angle is formed when two lines (or rays or line-segments) meet.
  • We learnt different types of angles,
    • Acute Angle- An angle that is less than 90 degree.
    • Right Angle- An angle that is 90 degree exactly.
    • Obtuse Angle- An angle that is greater than 90 degree but less than 180 degree.
    • Straight Angle- An angle that is 180 degree exactly.
    • Reflex Angle- An angle that is greater than 180 degree.
    • Two complementary angles- Measures add up to 90 degree.
    • Two supplementary angles- Measures add up to 180 degree.
    • Two adjacent angles- Have a common vertex and a common arm but no common interior.
    • Linear pair – Angles are Adjacent and supplementary.
    • Vertically Opposite Angles- Angles opposite to each other and are equal.

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