Grade IX - Mathematics

Author:
Rahul Chandra Saha
M.Sc., Mathematics

Lines and Pair of Angles​

Lines

Intersecting lines and non-intersecting lines
Two lines are called intersecting if they have a common point in their path. Otherwise they are called parellel lines.
Or, two lines are parellel if they touches each other at infinity.

Perpendicular lines
Two intersecting lines are called perpendicular to each other if the angle between them is 90°.
CF and DE are perpendicular lines.

Distance between two parallel lines
The distance or shortest distance between two parallel lines is the length of the perpendicular (line) segment between them. It does not matter from which point you draw the perpendicular segment

Distance between 2 parallel lines

Here, $\overline{AB}$ is the distance between the lines l and m.

Relation between Pair of Angles.

In the previous section we have learned about different pair of angles e.g. Complementary angles, supplementary angles, linear pair of angles. Here we try to find some relation between them.

Linear Pair Axioms

Axiom 1. If a ray stands on a line, then the sum of the two adjacent angles so formed is 180°.

Axiom 2. If the sum of the two adjacent angles 180° then the non-common arms of the angles form a line.

Theorem 1. If two lines intersect each other then the vertically opposite angles are equal.

Proof:

Let $\overline{AB}$ and $\overleftrightarrow{CD}$ be two straight lines intersecting at the point $E .$ According to the figure $\angle A E D$ and $\angle C E B$ and $\angle A E C$ and $\angle B E D$ are pair wise vertically opposite angles.
We have to prove that, $\angle A E D=\angle C E B$ and $\angle A E C=\angle B E D$
Now, $E$ is the source and $\overrightarrow{E C}$ is a ray. Then according to the Linear Pair axioms $\angle A E C+\angle C E B=180^{\circ}$.

Similar arguments ,assuming $\overrightarrow{E B}$ as ray, gives $\angle C E B+\angle B E D=180^{\circ} .$

Therefore, we get,

$\angle A E C+\angle C E B=\angle C E B+\angle B E D$

or $, \angle A E C=\angle B E D$

Similarly we can show that,\[\angle A E D=\angle C E B . \text { (Q.E.D.) }\]