Angles

Names of Angles

As the angle increases, the name changes:

Supplementary Angles

Two angles are supplementary when they add up to 180 degrees. They don’t have to be next to each other, just so long as the total is 180 degrees.

Examples:

60^o and 120^o are supplementary angles.
93^o and 87^o are supplementary angles.

Complementary Angles

Two Angles are Complementary when they add up to 90 degrees (a Right Angle).
They don’t have to be next to each other, just so long as the total is 90 degrees.

Examples:

60^o and 30^o are complementary angles.
5^o and 85^o are complementary angles.

CLASS ACTIVITY

1. Find the angle that is complementary to (a) 75^o  (b) 105^o.
2. Classify each of the following angles into its appropriate type: (a) 125^o (b) 198^o (c) 86^o

Measurement of Angles

The protractor is a mathematical instrument used for measuring and drawing angles. Angles are measured in degrees.

A protractor may be semicircular (i.e. 180 protractor) or circular (i.e. 360 protractor) in shape. There are two types of scales shown on a protractor, one is clockwise scale and the other is anticlockwise scale as shown above.

Example:

Measure angle AOB with your protractor.

Solution:

1. Place the centre point, O, of the protractor on the vertex (i.e. where the two arms of the angle meet) in such a way that the zero line of the protractor coincides with line OA of the angle. You may need to extend lines OA and OB.
2. Count round the numbers from point B as shown above.
3. Read off the measurement from the inner scale to obtain 20 .

CLASS ACTIVITY

1. Draw an angle 110 with a protractor.
2. Write down the sizes of these angles.

Identification of Angles

When two lines meet at a point, they form an angle. An angle is defined as the amount that one  line turns through to meet the other line.

The point B where two lines AB and CB meet is called the vertex. Lines AB and BC are called the arms of the angle. If the direction (turning) of a line is same as the direction of a clock then such rotation is called clockwise rotation. If the direction is in the opposite direction it is called as anti-clockwise or counter -clockwise rotation. In the drawing above, the angle at point B can be expressed as ABˆC or Bˆ or ∠ABC or ∠CBA

(Teacher to explain these notations)

Properties of Angles

Definitions

1. Any line drawn across set of parallel lines is referred to as a transversal.
2. Angles are said to be supplementary if their sum gives two right angles
3. Angles are said to be complementary if their sum gives one right angle

When a transversal cuts across a set of parallel lines we have the following three principles or laws of angles in display: corresponding angles, alternate angles, co-interior or allied angles.

Note: Corresponding angles are equal. Alternate angles are equal, but co-interior or allied angles are supplementary. These three laws require parallelism.      

Vertically Opposite Angles 

When two straight lines intersect, they form four angles; two angles opposite to each other are said to be vertically opposite.

Vertically opposite angles are equal. Hence, ∠XOV = ∠UOY; ∠XOU = ∠VOY.

CLASS ACTIVITY

1. Use your protractor to measure the following angles; ∠XOU, ∠XOV, ∠UOY and ∠VOY.
2. What do you notice?

Adjacent Angles on a Straight Line

In the diagram below, ∠XOY is a straight line, ∠XOZ and ∠YOZ lie next to each other, and they are referred to as adjacent angles on a straight line. In other words, when two angles lie beside each other and have a common vertex, we say they are adjacent to each other.

Since the sum of angles on a straight line is 180⁰,  ∠XOZ + ∠YOZ = 180^o. i.e.  a + b = 180^o

The sum of adjacent angles on a straight line is 180^o.

CLASS ACTIVITY

1. Use your protractor to measure angles labeled a and b.
2. Add angles a and b. What do you notice?

NOTE: Adjacent angles are said to be supplementary.

Alternate Angles

Alternate angles are equal

In the diagram drawn above a is alternate to b and r is alternate to s.

CLASS ACTIVITY

1. Use your protractor to measure the angles labeled a, b, r and s.
2. What is your observation?

Corresponding Angles

Corresponding angles are equal.

CLASS ACTIVITY

Use your protractor to measure the angles labeled x and y and then p and q in the in the diagram above. What do you notice?

NOTE: Angles x and y are called corresponding angles. Also angles p and q are called corresponding angles.

Therefore, when a transversal cut parallel lines corresponding angles formed are equal.

Identification of Angles at a Point

The sum of angles at a point is 360^o.

In the diagram below all the lines intersect at a point O.

CLASS ACTIVITY

1. Use your protractor to measure angles labeled p, q, r and s.
2. Add angles p, q, r and s. What is your observation?

Note: From the activity above, your result should add up to 360⁰.

Identification of Angles on a Straight Line

When a straight line stands on another straight line, two adjacent angles are formed. The sum of two adjacent angles in the case shown above is 180^o.

Worked Example:

Find the value of the unknown angle in the diagram below

Solution

35^o + p = 180^o (angles on a straight line)

p = 180^o − 35^o

p = 145^o

CLASS ACTIVITY

1. What do you understand by the following principles?

• Corresponding angles
• Alternate angles
• Co-Interior or Allied angles
• Vertically opposite angles
• Sum of angles on a straight line
• Angles at a point.

2. Make rough sketches to explain them.

The Sum of Angles in a Triangle

The sum of angles in a triangle is 180⁰. To show this, draw line LM through the top vertex of the triangle, parallel to the base BC. Label each angle as shown in the diagram below.

From the above diagram:

b = d   (alternate angles)

c = e    (alternate angles)

But  d + a + e = 180^o   (sum of angles on a straight line)

Since d + e  = b + c

Hence: a + b + c = d + a + e = 180^o

Thus, the sum of angles of a triangle = 180^o.

Worked Example:

Find the size of angle x in this triangle.

Solution:

x + 64^o + 88^o = 180^o (sum of angles in a triangle)

x + 152^o= 180^o

x = 180^o – 152^o = 28^o

PRACTICE QUESTIONS

1. Find the complement and supplement of the following angles: (i) 63^o(ii) 124^o(iii) 135^o (iv) 13^o (v) 154^o (vi) 28^o (vii) 32^o (viii) 51^o
2. If m∠2 = 42^o and ∠1 and ∠2 are complementary angles, find m∠1.
3. If m∠1 = 90^oand ∠1 and ∠2 form a linear pair, find m∠2.
4. If ∠1 and ∠2 are supplementary angles and m∠2 = 131^o, find m∠1.
5. If ∠1 and ∠2 form a right angle and m∠1 = 55^o, find m∠2.