Adding and Subtracting Angles

The measures of anglesthey are not always whole quantities, we can have, for example, an angle measuring between 90° and 91°. For these cases, submultiples of the degree are used.

Operations between angle measures, such as addition and subtraction, can involve such submultiples. Therefore, it is necessary to understand what they are and how they are related.

Submultiples of degree

You submultiples of degree are minutes and seconds, these two units express quantities less than a degree.

The degree, minutes, and seconds are related as follows:

• 1 degree = 60 minutes
• 1 minute = 60 seconds

It is common to use symbols for degrees (°), minutes (‘) and seconds (“). So, equivalently, we have:

• 1° = 60′
• 1′ = 60″

Example: Use submultiples of degrees to express a 45.5° angle.

45.5° is the angle that is exactly in the middle of the angles 45° and 46°, that is, it is 45° plus half a degree.

As 1 degree is 60 minutes, so half of 1 degree is 30 minutes:

1° = 60′ ⇒ 0,5° = 30′

Therefore, 45.5° = 45°30′.

It reads: 45 degrees and 30 minutes.

Adding Angles

to do the adding angles, we add seconds to seconds, minutes to minutes, and degrees to degrees. Then we simplify the results. If after addition we have:

• 60 seconds or more ⇒ convert to minutes;
• 60 minutes or more ⇒ convert to degrees.

Example: Add angle measures.

a) 35° 20′ 10″ + 15° 30′ 8″

35° 20′ 10″
+15° 30′ 8″
,,50° 50′ 18″

Therefore:

35° 20′ 10″ + 15° 30′ 8″ = 50° 50′ 18″

b) 90° 60′ + 5° 70′ 85″

,,90° 60′ 00″
+5° 70′ 85″
95° 130′ 85″

In this case, we must simplify the result.

We always start with seconds: 95° 130′ 85″

85″ = 60″ + 15″ = 1′ + 15″ = 1′ 15″ ⇒95° 130′ 85″ = 95° 131′ 15″

Now, let's go to the minutes: 95°131′15″

131′ = 60′ + 60′ + 10′ = 1° + 1° + 10′ = 2°10′ ⇒ 95°131’15” = 97°10’15”

Therefore:

90° 60′ + 5° 70′ 85″ = 97° 10′ 15″

angle subtraction

to do the angle subtraction, we subtract seconds from seconds, minutes from minutes, and degrees from degrees.

Whenever it is necessary to “borrow”, we must remember the relations between the submultiples.

Example: Calculate subtractions between angle measures.

a) 40° 28′ 12″ – 10° 13′ 6″

,,40° 28′ 12″
-10° 13′ 6″
,30° 15′ 6″

Therefore:

40° 28′ 12″ – 10° 13′ 6″ = 30° 15′ 6″
,,

b) 90° 25′ – 75° 20′ 30″

,,90° 25′ 00″
-75° 20′ 30″

?

We cannot subtract 30 from 0. In this case, we need to “borrow” the minute place.

1′ = 60″ ⇒ the second place will be borrowed 1′, which corresponds to borrowed 60″.

,,90° 24′ 60″
-75° 20′ 30″
,15° 4′ 30″

Therefore:

90° 25′ – 75° 20′ 30″ = 15° 4′ 30″

You may also be interested:

• Vertex Opposite Angles
• Complementary and Supplementary Angles
• Interior and exterior side angles