# ANGLES: Finding Unknown Angles

*Angles on a straight line*

*Angles on a straight line*

*Example:*

Consider a straight line ABC. Draw a line DB on to it meeting it at B.

Now, using a protractor, measure ∠x ∠y.

In our case,

∠x = 80°

∠y = 100°

∠x + ∠y = 80° + 100° = 180°

*Take another example:*

Consider the straight line ABC below. Drop a few lines onto point B and, using a protractor, measure all the angles formed.

In our case below, we have dropped 2 lines DB and EB, and the angles formed by them are as follows:

∠x = 45°

∠y = 63°

∠z = 72°

∠x + ∠y + ∠z = 45° + 63° + 72° = 180°

From the above examples it is clear that thesum of angles on a straight line is 180°.

*Angles at a point*

*Angles at a point*

Consider 2 straight lines AB and CD crossing at point O.

We know that,

∠w + ∠x = 180° (sum of angles on a straight line is 180°)

Similarly,

∠y + ∠z is 180° (sum of angles on a straight line is 180°)

Adding all the angles at O, we get,

∠w + ∠x + ∠y + ∠z = 180° + 180° = 360°

*Take another example:*

Mark a point O and extend a few lines in any direction from it as below.

Now using a protractor measure all the angles at point O.

In our case,

∠v = 120°

∠w = 70°

∠x = 20°

∠y = 90°

∠z = 60°

∠v + ∠w + ∠x + ∠y + ∠z = 120° + 70° +20 ° + 90° + 60° = 360°

It is clear from the above examples that thesum of all angles at a point is 360°.

*Vertically opposite angles*

*Vertically opposite angles*

Consider 2 straight lines AB and CD crossing each other.

Now, using a protractor measure ∠m, ∠x, ∠n and ∠y.

∠m and ∠n form one pair of vertically opposite angles.In our case,

∠x and ∠y form a second pair of vertically opposite angles.

∠m = 120°

∠x = 60°

∠n = 120°

∠y = 60°

What we notice is,

∠m = ∠n, and

∠x = ∠y

Therefore, vertically opposite angles are equal to each other.

**Up for some challenge?**

Following are 3 straight lines crossing each other at O.

Can you mark the three pairs of vertically opposite angles?

[Help: You may draw similar lines on a piece of paper and use a protractor to measure the angles.]

Answer:

The 3 pairs of vertically opposite angles are:

1. ∠a and ∠d

2. ∠b and ∠e

3. ∠c and ∠f