Angles and their calculation encompass basic geometric concepts: the sum of angles in a triangle is 180°, and in a quadrilateral it is 360°.

Key operations include adding/subtracting degrees, minutes, and seconds, as well as using the relationships of complementary (90°) and supplementary (180°) angles. An exterior angle is always supplementary to the interior angle.

Basic rules for calculating angles:
Triangle: The sum of the interior angles is 180°.
Quadrilateral: The sum of the interior angles is 360°.
Exterior angles: The sum of the exterior angles of a polygon is always 360°.
Angles with parallel sides: They can be equal or supplementary (180°).

Examples of calculations:
Unknown angle in a quadrilateral: If three angles are known, the fourth is calculated as 360° minus the sum of the three known angles.

Problem No 1:
Solve the angle at point C of triangle ABC, where the interior angle at point A is 30 degrees, and the exterior angle at point B is 135 degrees.

Solution:

• Let the interior angles of triangle ABC be denoted as ∠A, ∠B, and ∠C.
• Given: ∠A = 30°
• The exterior angle at point B is 135°. Recall that the exterior angle and the interior angle at the same vertex are supplementary, meaning:
  ∠B + exterior angle at B = 180°
• Therefore,
  ∠B = 180° − 135° = 45°
• Since the sum of interior angles in a triangle is 180°, we have:
  ∠A + ∠B + ∠C = 180°
• Substitute the known values:
  30° + 45° + ∠C = 180°
• Calculate ∠C:
  ∠C = 180° − 30° − 45° = 105°

Answer:
The angle at point C is 105 degrees.

Adding angles: Add degrees, minutes, and seconds accordingly (carrying over when minutes or seconds exceed 60).

Given:
127 degrees 35 minutes and 45 seconds
We want to add 12 degrees 45 minutes and 41 seconds

127° 35′ 45”

+

12° 45′ 41”

=

139° (80=60+20)’ (86=60+26)”

which gives

140° 21′ 26”

KEEP IN MIND

If minutes exceed 60, convert to degrees. If seconds exceed 60, convert to minutes

For example, if minutes = 65:

65 minutes = 1 degree and 5 minutes

86 seconds = 1 minute and 26 seconds

Substracting angles:

Step 1: Write down the angles to be subtracted.
135 degrees 0 minutes 0 seconds
36 degrees 15 minutes 5 seconds

Step 2: Subtract the seconds.
0 seconds – 5 seconds: Since 0 is less than 5, borrow 1 minute (which is 60 seconds) from the minutes column.
Minutes after borrowing: 0 – 1 = -1 (we will fix this in the next step)
Seconds after borrowing: 0 + 60 = 60 seconds
Now subtract seconds: 60 – 5 = 55 seconds

Step 3: Subtract the minutes.
Now minutes are 59 (60 borrowed 1 by degrees and -1 after borrowing to seconds in last step).
Degrees after borrowing: 135 – 1 = 134 degrees
Minutes after calculating: 59 – 15 = 44 minutes

Step 4: Subtract the degrees.
134 degrees – 36 degrees = 98 degrees

Step 5: Final result.
Degrees: 98
Minutes: 44
Seconds: 55

Answer: 98 degrees 44 minutes 55 seconds

KEEP IN MIND

When we have to subtract from 0, we have to “borrow.”

One borrowed minute to seconds is worth 60. One borrowed degree to minutes is 60!