ANGLES: Finding Unknown Angles

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 the sum of angles on a straight line is 180°.

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 the sum of all angles at a point is 360°.

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.

∠x and ∠y form a second pair of vertically opposite angles.
In our case,
∠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