Module 17: Angles

You will need a calculator near the end of this module.

an angle grinder

Angle measurement is important in construction, surveying, physical therapy, and many other fields. We can visualize an angle as the figure formed when two line segments share a common endpoint. We can also think about an angle as a measure of rotation. A full rotation or a full circle is 360^\circ, so a half rotation or U-turn is 180^\circ, and a quarter turn is 90^\circ.

We often classify angles by their size.

Acute Angle: between 0^\circ and 90^\circ

Right Angle: exactly 90^\circ

Obtuse Angle: between 90^\circ and 180^\circ

Straight Angle: exactly 180^\circ

Reflexive Angle: between 180^\circ and 360^\circ

Lines that form a 90^\circ angle are called perpendicular. As shown below, the needle should be perpendicular to the body surface for an intramuscular injection.


Identify each angle shown below as acute, right, obtuse, straight, or reflexive.






Find the measure of each unknown angle.


Angles in Triangles

If you need to find the measures of the angles in a triangle, there are a few rules that can help.

The sum of the angles of every triangle is 180^\circ.

If any sides of a triangle have equal lengths, then the angles opposite those sides will have equal measures.


Find the measures of the unknown angles in each triangle.






Angles and Parallel Lines

Two lines that point in the exact same direction and will never cross are called parallel lines. If two parallel lines are crossed by a third line, sets of equally-sized angles will be formed, as shown in the following diagram. All four acute angles will be equal in measure, all four obtuse angles will be equal in measure, and any acute angle and obtuse angle will have a combined measure of 180^\circ.


12. Find the measures of angles A, B, and C.

Degrees, Minutes, Seconds

It is possible to have angle measures that are not a whole number of degrees. It is common to use decimals in these situations, but the older method—called the degrees-minutes-seconds or DMS system—divides a degree using fractions out of 60: a minute is \frac{1}{60} of a degree, and a second is \frac{1}{60} of a minute, which means a second is \frac{1}{3,600} of a degree. (Fortunately, these conversions work exactly like time; think of 1 degree as 1 hour.) For example, 2.5^\circ=2^\circ30'.

We will look at the procedure for converting between systems, but there are online calculators such as the one at which will do the conversions for you.

If you have latitude and longitude in DMS, like N 18^\circ54'40'' W 155^\circ40'51'', and need to convert it to decimal degrees, the process is fairly simple with a calculator.

Converting from DMS to Decimal Degrees

Enter \text{degrees}+\text{minutes}\div60+\text{seconds}\div3600} in your calculator. Round the result to the fourth decimal place, if necessary.[1]


Convert each angle measurement from degrees-minutes-seconds into decimal form. Round to the nearest ten-thousandth, if necessary.

13. 67^\circ48'54''

14. 19^\circ37'25''

15. 34^\circ14'12''

Going from decimal degrees to DMS is a more complicated process.

Converting from Decimal Degrees to DMS

  1. The whole-number part of the angle measurement gives the number of degrees.
  2. Multiply the decimal part by 60. The whole number part of this result is the number of minutes.
  3. Multiply the decimal part of the minutes by 60. This gives the number of seconds (including any decimal part of seconds).

For example, let’s convert 15.374^\circ.

  1. The degrees part of our answer will be 15.
  2. The decimal part times 60 is 0.374\cdot60=22.44 minutes. The minutes part of our answer will be 22.
  3. The decimal part times 60 is 0.44\cdot60=26.4 seconds. The seconds part of our answer will be 26.4.

So 15.374^\circ=15^\circ22'26.4''.


Convert each angle measurement from decimal into degrees-minutes-seconds form.

16. 26.785^\circ

17. 58.216^\circ

18. 41.13^\circ


  1. We round to four decimal places because 1 second of angle is \frac{1}{3,600} of a degree. This is a smaller fraction than \frac{1}{1,000} so our precision is slightly better than the thousandths place.


Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Technical Mathematics Copyright © 2020 by Morgan Chase is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.