Contents
Complementary Angles:
When the sum of the measures of two angles is 90°, the angles are called complementary angles.
Supplementary Angles:
Do you notice that the sum of the measures of the angles in each of the above pairs comes out to be 180º? Such pairs of angles are called supplementary angles.
When two angles are supplementary, each angle is said to be the supplement of the other.
When the sum of the measures of two angles is 180°, the angles are called supplementary angles.
Adjacent Angles:
When you open a book it looks like the above figure. In A and B, we find a pair of angles, placed next to each other.
Look at this steering wheel of a car. At the centre of the wheel you find three angles being formed, lying next to one another.
At both the vertices A and B, we find, a pair of angles are placed next to each other.
These angles are such that:
1) they have a common vertex;
2) they have a common arm; and
3) the non-common arms are on either side of the common arm.
Such pairs of angles are called adjacent angles. Adjacent angles have a common vertex and a common arm but no common interior points.
Linear Pair:
A linear pair is a pair of adjacent angles whose non-common sides are opposite rays.
In fig above, observe that the opposite rays (which are the non-common sides of \(\angle{1}\) and \(\angle{2}\) form a line. Thus, \(\angle{1} + \angle{2}\) amounts to 180°.
Have you noticed models of a linear pair in your environment?
Note carefully that a pair of supplementary angles form a linear pair when placed adjacent to each other. Do you find examples of linear pair in your daily life?
Vertically Opposite Angles:
Next take two pencils and tie them with the help of a rubber band at the middle.
Look at the four angles formed \(\angle{1}\), \(\angle{2}\), \(\angle{3}\) and \(\angle{4}\).
\(\angle{1}\) is vertically opposite to \(\angle{3}\).
and \(\angle{2}\) is vertically opposite to \(\angle{4}\).
We call \(\angle{1}\) and \(\angle{3}\), a pair of vertically opposite angles.
Draw two lines l and m, intersecting at a point. You can now mark \(\angle{1}\), \(\angle{2}\), \(\angle{3}\) and \(\angle{4}\) as in the fig.
Take a trace copy of the figure on a transparent sheet.
Place the copy on the original such that \(\angle{1}\) matches with its copy, \(\angle{2}\) matches with its copy, … etc.
Fix a pin at the point of intersection. Rotate the copy by 180°. Do the lines coincide again?
You find that \(\angle{1}\) and \(\angle{3}\) have interchanged their positions and so have \(\angle{2}\) and \(\angle{4}\).
This has been done without disturbing the position of the lines.
Thus, \(\angle{1}\) = \(\angle{3}\) and \(\angle{2}\) = \(\angle{4}\).
We conclude that when two lines intersect, the vertically opposite angles so formed are equal.
Let us try to prove this using Geometrical idea. Let us consider two lines l and m.
We can arrive at this result through logical reasoning as follows:
Let l and m be two lines, which intersect at O, making angles \(\angle{1}\), \(\angle{2}\), \(\angle{3}\) and \(\angle{4}\).
We want to prove that \(\angle{1}\) = \(\angle{3}\) and \(\angle{2}\) = \(\angle{4}\)
Now, \(\angle{1}\) = 180º – \(\angle{2}\) (Because \(\angle{1}\), \(\angle{2}\) form a linear pair, so, \(\angle{1}\) + \(\angle{2}\) = 180°.
Similarly, \(\angle{3}\) = 180º – \(\angle{2}\) (Since \(\angle{2}\), \(\angle{3}\) form a linear pair, so,\(\angle{2}\) + \(\angle{3}\) = 180°.
Therefore,\(\angle{1}\) = \(\angle{3}\)
Similarly, we can prove that \(\angle{2}\) = \(\angle{4}\)
Transversal:
You might have seen a road crossing two or more roads or a railway line crossing several other lines. These give an idea of a transversal.
A line that intersects two or more lines at distinct points is called a transversal.
For example, in the following fig. p is a transversal to the lines l and m.
Now, p is not a transversal, although it cuts two lines l and m, because it does not intersect the lines at different points. In fact, it intersects two lines at the same point.
Angles made by a transversal with two lines:
Let l and m be two lines and n be a transversal intersecting them at P and Q respectively.
Clearly, lines l, m and n make eight angles, four at P and the remaining four at Q. We label them 1 to 8 for the sake of convenience and classify them in the following groups:
Exterior Angles:
The angles whose arms do not include the line segment PQ are called exterior angles.
In the above fig., angles 1, 2, 7 and 8 are exterior angles.
Interior Angles:
The angles whose arms include line segment PQ are called interior angles.
In the above fig., angles 3, 4, 5 and 6 are interior angles.
Corresponding Angles:
A pair of angles in which one arm of both the angles is on the same side of the transversal and their other arms are directed in the same sense is called a pair of corresponding angles.
In the above fig., \(\angle{1}, \angle{5}\); \(\angle{2}, \angle{6}\); \(\angle{3}, \angle{7}\) and \(\angle{4}, \angle{8}\) are four pairs of corresponding angles.
Alternate Interior Angles:
A pair of angles in which one arm of each of the angle is on opposite sides of the transversal and whose other arms include segment PQ is called a pair of alternate interior angles.
In fig. above, \(\angle{3}\) and \(\angle{5}\), \(\angle{4}\) and \(\angle{6}\) form pairs of alternate interior angles.
Alternate Exterior Angles:
A pair of angles in which one arm of each of the angle is on opposite sides of the transversal and whose other arms are directed in opposite direction and do not include segment PQ is called a pair of alternate exterior angles.
In fig. above, \(\angle{2}\) and \(\angle{8}\), \(\angle{4}\) and \(\angle{7}\) form pairs of alternate exterior angles.