1. Intersecting Lines:
When two straight lines meet at a single point on a flat plane, they are called intersecting lines. The point where they meet is the point of intersection.
2. Linear Pairs:
Angles that lie next to each other on a straight line and share a common arm are called a linear pair.
Key Property: The angles in a linear pair add up to 180° (they are supplementary).
Example:
If \(x\) and \(y\) are the measures of linear pair of angles, then find the value of \(y\) given \(x\) \(=\) \(65^{\circ}\).
Solution:
Given that, \(x\) \(=\) \(65^{\circ}\).
By the property of linear pair of angles, \(x\) \(+\) \(y\) \(=\) \(180^{\circ}\).
\(\Rightarrow 65^{\circ} + y = 180^{\circ}\)
\(\Rightarrow y = 180^{\circ} - 65^{\circ}\)
\(\Rightarrow y = 115^{\circ}\)
Some real life examples:
The following are some real-life examples where we can observe linear pair of angles.
- The ladder makes a linear pair of angles with the ground.
- The writing quill in the inked dip makes a linear pair of angle with the table.
- The knife makes a linear pair of angle with the chopping board.
3. Vertically Opposite Angles:
When two lines intersect, they form four angles. The pairs of angles that are opposite to each other across the vertex are called vertically opposite angles.
Key Property: Vertically opposite angles are always equal to each other.
If lines \(l\) and \(m\) intersect, then the angle on the left equals the angle on the right, and the angle on top equals the angle on the bottom.
Illustration:
In the figure, the lines \(p\) and \(q\) intersect at \(O\) forming four pair of angles \(a\), \(b\), \(c\) and \(d\).
The angle \(a\) is vertically opposite to \(c\), and the angle \(b\) is vertically opposite to \(d\) and vice versa.
Here \(\angle a\) \(=\) \(\angle c\) and \(\angle b\) \(=\) \(\angle d\).
Let us prove the above equivalence as follows:
Consider \(\angle a\) and \(\angle b\).
These two angles form a linear pair. Hence by the property of linear pair of angles \(\angle a + \angle b = 180^{\circ}\).
\(\Rightarrow\) \(\angle a\) \(=\) \(180^{\circ}\) \(-\) \(\angle b\) ……\((1)\)
Also, \(\angle b\) \(=\) \(180^{\circ}\) \(-\) \(\angle a\) ……\((2)\)
Consider \(\angle b\) and \(\angle c\).
These two angles form a linear pair. Hence by the property of linear pair of angles \(\angle b + \angle c = 180^{\circ}\).
\(\Rightarrow\) \(\angle c\) \(=\) \(180^{\circ}\) \(-\) \(\angle b\) ……\((3)\)
Thus from equations \((1)\) and \((3)\) we have:
\(\angle a\) \(=\) \(\angle c\)
Similarly, consider \(\angle a\) and \(\angle d\).
These two angles form a linear pair. Hence by the property of linear pair of angles \(\angle a + \angle d = 180^{\circ}\).
\(\Rightarrow\) \(\angle d\) \(=\) \(180^{\circ}\) \(-\) \(\angle a\) ……\((4)\)
And from equations \((2)\) and \((4)\) we have:
\(\angle b\) \(=\) \(\angle d\)
Therefore, the vertically opposite angles formed by the lines \(p\) and \(q\) at the point \(O\) are equal in measure.
Example:
Find the unknown angle \(b\) in the figure.
Solution:
From the figure, we observe that the \(\angle SOU\) and \(\angle TOV\) are vertically opposite angles formed by the line segments \(ST\) and \(UV\).
By definition, the vertically opposite angles are equal.
Thus \(\angle SOU\) \(=\) \(\angle TOV\).
\(\Rightarrow\) \(b\) \(=\) \(50^{\circ}\)
Therefore, the unknown angle \(b\) is \(50^{\circ}\)
4. Perpendicular Lines:
When two lines intersect each other in such a way that the angles formed between them are all \(90^\circ\) (right angles), the lines are said to be perpendicular lines.
If all four angles at the intersection point are equal, each must be exactly \(90^\circ\) because the full rotation around a point is \(360^\circ\) (\(\frac{360^\circ}{4} = 90^\circ\)).
Symbolically, if line \(l\) is perpendicular to line \(m\), it is written as \(l \perp m\).
5. Parallel Lines:
When two or more lines lie on the same plane surface but never intersect each other, no matter how far they are extended in either direction, they are called parallel lines.
Key Property: The perpendicular distance between two parallel lines remains constant throughout their entire length.Symbolically, if line \(l\) is parallel to line \(m\), it is written as \(l \parallel m\).