Introduction to Angles

When two lines intersect at a point, the measure of the “opening” between these two lines is called an angle. Angles are usually measured in degrees.

Ever seen a game of football? Noticed where the players take the corner from? Well, the point at which the lines intersect is what forms an angle!

The Big Idea: What are angles?

Before getting into angles, let us understand what lines and rays are. We know that a collection of points is called a line. Imagine that there is only one fixed point at one end of a line, while the other end extends indefinitely. Such a line is called a ray. A ray has one start point, but the other end extends infinitely. Usually, the infinite end is represented by an arrowhead.

As was said earlier, an angle is simply the opening between two intersecting rays. Look at the figure below to get a clearer picture of what that means.

Types of Angles

Six types of angles can be formed by two-line segments meeting each other at a point or intersecting each other. They are:

  • Acute Angle: When the angle is greater than 0° but lesser than 90°.
  • Obtuse Angle: When the angle is more than 90° but less than 180°.
  • Reflex Angle: When the angle is more than 180° but less than 360°.
  • Right Angle: When the angle is equal to 90°.
  • Straight Angle: When the angle is equal to 180°.
  • Full Angle: When the angle is equal to 360°.

Vertically Opposite Angles

When two lines intersect each other in an ‘\({x}\)’ shape, four angles are formed. Take a look at the picture above, in which lines PQ and RS intersect each other. The four angles formed are marked as \({a,\,n,\,b,\text{ and }m}\). The pairs of these angles which are marked in same colours (yellow and blue) are called vertically opposite angles. So, angles b and a are vertically opposite angles, while m and n are also vertically opposite angles. Now the rules of geometry state that for any two straight lines which intersect each other at any angle, both sets of vertically opposite angles so formed are equal.

Complementary and Supplementary Angles

Before we can understand supplementary and complementary angles, let us look at a straight angle. 

The term “straight angle” originates from the straight line. As we all know, a straight line lies flat and so if it were to be counted as 2 rays starting from the same point, moving in different directions, the “open space” between them would measure 180°. Now, you add a third ray that starts from the same point as the two rays that made the straight line, but in a different direction from the first two (as in the figure below). Then the single 180° angle gets split into 2 angles of different measure. However, they will still add up to 180°. Therefore, such a pair of angles which add up to 180° are called supplementary angles.

Similarly, if a pair of angles adds up to 90°, those two angles are called complementary angles. Some examples of complementary angles are 35° and 55°, 32.5° and 57.5° and 45° and 45°. All of these pairs add up to 90°.

Parallel Lines and their angles

We discussed several scenarios in which two-line segments meet or intersect to create angles of different kinds. Now let us see what happens when one straight line intersects two other parallel straight lines. When a straight-line crosses two other parallel straight lines, it will meet two intersection points, one for each of the two lines. The angles which have the same relative position at both the intersection points are called corresponding angles. All the pairs of corresponding angles are also equal to each other. The exact opposite of the corresponding angles is called alternate interior angles.

 

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