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# Revision Notes on Lines and Angles

Point

A point is a geometrical element which has no dimensions. Line

A line is a straight path which has no endpoints. Line Segment

A line segment is a straight path which has two endpoints. Ray

A ray is a line which has one endpoint and endless from another side. ## Angles

The corners made by the intersection of two lines or line segments are called Angles. We write angle as ∠ABC in first figure and ∠XOY, ∠ZOW, ∠YOW and ∠XOZ are angles in the second figure.

## Related Angles

### 1. Complementary Angles

If the sum of two angles is 90° then they are said to be complementary angles. Or you can say that two angles which make up a right angle are called Complementary Angle.

### 2. Supplementary Angles

If the sum of two angles is 180° then they are said to be supplementary angles. If two angles are supplementary then they are the supplement to each other. ### 3. Adjacent Angles

It is the pair of two angles which are placed next to each other.

Adjacent angles have-

• A common vertex.

• A common arm.

• A non-common arm could be on either side of the common arm. ### 4. Linear Pair

A pair of adjacent angles whose non-common arm makes a single line i.e. they are the opposite rays.

A linear pair is also a pair of supplementary angles as their sum is 180°. The above pair of angles is –

• Adjacent, as they have one common arm.

• Supplementary, as the sum of two angles, is 180°.

• The linear pair, as the sum is 180° and the non – common arms are opposite rays.

### 5. Vertically Opposite Angles

When two lines intersect each other then they form four angles. So that

• ∠a and ∠b is pair of vertically opposite angles.

• ∠n and ∠m is pair of vertically opposite angles. Vertically opposite angles are equal.

## Pairs of Lines

### 1. Intersecting Lines

If two lines touch each other in such a way that there is a point in common then these lines are called intersecting lines.

That common point is called a Point of Intersection. Here, line l and m intersect each other at point C.

### 2. Transversal

If a line intersects two or more lines at different points then that line is called Transversal Line. ### 3. Angles made by a transversal

When a transversal intersects two lines then they make 8 angles. Some of the angles made by transversal-

 Types of Angles Angles shown in figure Interior Angles ∠6, ∠5, ∠4, ∠3 Exterior Angles ∠7,∠8,∠1,∠2 Pairs of Corresponding Angles ∠1 and ∠5,∠2 and ∠6, ∠3 and ∠7,∠4 and ∠8 Pairs of Alternate Interior Angles ∠3 and ∠6,∠4 and ∠5 Pairs of Alternate Exterior Angles ∠1 and ∠8,∠2 and ∠7 Pairs of Interior Angles on the same side of the transversal ∠3 and ∠5,∠4 and ∠6

## Transversal of Parallel Lines

The two lines which never meet with each other are called Parallel Lines. If we have a transversal on two parallel lines then- a. All the pairs of corresponding angles are equal.

∠3 = ∠7

∠4 = ∠8

∠1 = ∠5

∠2 = ∠6

b. All the pairs of alternate interior angles are equal.

∠3 = ∠6

∠4 = ∠5

c. The two Interior angles which are on the same side of the transversal will always be supplementary.

∠3 + ∠5 = 180°

∠4 + ∠6 = 180°

### Checking for Parallel Lines

This is the inverse of the above properties of the transversal of parallel lines.

• If a transversal passes through two lines so that the pairs of corresponding angles are equal, then these two lines must be parallel.

• If a transversal passes through two lines in so that the pairs of alternate interior angles are equal, then these two lines must be parallel.

• If a transversal passes through two lines so that the pairs of interior angles on the same side of the transversal are supplementary, then these two lines must be parallel.

Example: 1

If AB ∥ PQ, Find ∠W. Solution: We have to draw a line CD parallel to AB and PQ passing through ∠W.

∠QPW = ∠PWC = 50° (Alternate Interior Angles)

∠BAW =∠CWA = 46°(Alternate Interior Angles)

∠PWA = ∠PWC +∠CWA

= 50°+ 46°= 96°

Example: 2

If XY ∥ QR with ∠4 = 50° and ∠5 = 45°, then find all the three angles of the ∆PQR. Solution:

Given:  XY ∥ QR

∠4 = 50° and ∠5 = 45°

To find: ∠1, ∠2 and ∠3

Calculation: ∠1 + ∠4 + ∠5 = 180° (sum of angles making a straight angle)

∠1 = 180°- 50°- 45°

∠1 = 85°

PQ is the transversal of XY and QR, so

∠4 = ∠2 (Alternate interior angles between parallel lines)

∠2 = 50°

Pr is also the transversal of XY and QR, so

∠5 = ∠3 (Alternate interior angles between parallel lines)

∠3 = 45°

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