Special Pairs of Angles 1

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We will now study properties of some angles which occur in pairs. These properties are used to prove many of the theorems in plane geometry. Chapter 167 represents these angles graphically.

Vertically Opposite Angles

When 2 lines intersect, they form two pairs of angles which are opposite to each other. Each pair is called a pair of Vertically Opposite angles. We can also intuitively see that in such a pair, each angle is equal to the other. This can be visualize easily by the movement of the arms of a scissor.

When one of the angles increases or decreases, the vertically opposite angle also increases or decreases such that both are always equal.

Adjacent Angles

Two angles are said to adjacent when the following conditions are met.

When 2 straight lines intersect, 2 pairs of Adjacent angles are formed. But these are a special case of adjacent angles which also add up to a straight angle.
 * 1) They have the same vertex
 * 2) One of their arms coincide with each other and
 * 3) The other 2 arms are on either side of the common arm Or the interiors of both the angles do not overlap.

Supplementary Angles

When the sum of any 2 angles is a Straight Angle (180 degrees), they are called Supplementary Angles.

We saw in the previous paragraph that when 2 straight lines intersect, 2 pairs of Adjacent angles are formed which are also supplementary.

Supplementary Angles need not be adjacent angles. In a complex figure, two angles which are situated physically apart can be called supplementary if their angle measures total to 180 degrees.

Linear Pair

If 2 angles are Supplementary as well as adjacent, they are also called a Linear Pair.

This is because it can be visualized as a straight line resting on another straight line forming 2 adjacent angles whose sum is a straight angle!

When 2 straight lines intersect, 4 angles are formed. If we know one of the angles all the other three angles can be found.

Complementary Angles

When the sum of any 2 angles is a Right Angle (90 degrees), they are called Complementary Angles.

Complementary Angles need not be adjacent angles. In a complex figure, two angles which are situated physically apart can be called complementary if their angle measures total to 90 degrees.

Identifying 2 physically separated angles as complementary or supplementary may help us logically prove certain geometrical statements.

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