## ANGLES: Finding Unknown Angles

#### ANGLES

This lesson covers the following:

- Sum of angles on a straight line
- Sum of angles at a point
- Vertically opposite angles

#### 1. Finding the sum of angles on a straight line

*Example 1:*
Consider a straight line ABC. Draw a line DB on to it meeting it at B.

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

In the figure above,

∠x = 80°

∠y = 100°

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

In the figure above,

∠x = 80°

∠y = 100°

**∠x + ∠y = 80° + 100° = 180°**

*Example 2:*
Consider the straight line ABC below. Drop a few lines onto point B and, using a protractor, measure all the angles formed.

In the figure above, we have dropped 2 lines DB and EB, and the angles formed by them are as follows:

∠x = 45°

∠y = 63°

∠z = 72°

In the figure above, 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 examples above it is clear that the

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

#### 2. Finding the sum of angles at a point

*Example 1:*
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,

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°**

*Example 2:*
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°

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°**From the examples above it is clear that the

**sum of angles at a point is 360°**.

#### 3. Vertically Opposite Angles

Consider 2 straight lines AB and CD crossing each other.

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

In our case,

∠m = 120°

∠x = 60°

∠n = 120°

∠y = 60°

In our case,

∠m = 120°

∠x = 60°

∠n = 120°

∠y = 60°

∠m and ∠n form one pair of vertically opposite angles.

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

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

What we notice is,

∠m = ∠n , and

∠x = ∠y

Therefore, vertically opposite angles are equal to each other.

∠m = ∠n , and

∠x = ∠y

Therefore, vertically opposite angles are equal to each other.

#### 4. 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.]

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:

The 3 pairs of vertically opposite angles are:

- ∠a and ∠d
- ∠b and ∠e
- ∠c and ∠f