17 Angles

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

Angle Measure

the legs of a woman wearing high heels
Photo by Rangga Cahya Nugraha on Unsplash

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. One 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 relative to these 90^\circ and 180^\circ benchmarks.

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

Exercises

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

  1. an angle marked with a small square; one ray pointing exactly southwest and the other ray pointing exactly southeast.
  2. an angle with one ray pointing west and the other ray pointing northeast. the interior of the angle on the north side is marked.
  3. an angle with one ray pointing west and the other ray pointing northeast. the interior of the angle on the south side is marked.
  4. an angle with one ray pointing west and the other ray pointing east. the interior of the angle on the north side is marked.
  5. an angle with one ray pointing east and the other ray pointing northeast. the interior of the angle is marked.

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.

diagram showing injection angles of four hypodermic syringes
Image by British Columbia Institute of Technology (BCIT) on Wikimedia Commons

When lines cross, they form angles. No surprises there. If we know the measure of one angle, we may be able to determine the measures of the remaining angles using a little logic.

Exercises

Find the measure of each unknown angle.

  1. two straight lines crossing to form an X; small angle at right labeled 53 degrees, large angle at top labeled a, small angle at left labeled b, large angle at bottom labeled c

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.

Exercises

Find the measures of the unknown angles in each triangle.

  1. right triangle with one angle 53 degrees; the other acute angle is unknown.

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.

Exercises

  1. 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—which is 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 units 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 https://www.fcc.gov/media/radio/dms-decimal 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]

Exercises

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

  1. 18^\circ54'40''
  2. 155^\circ40'51''
  3. 34^\circ11'32.5''

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 15.3740^\circ.

  1. The degrees part of our answer will be 15.
  2. The decimal part times 60 is 0.3740\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 15.3740^\circ=15^\circ22'26.4''.

Exercises

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

  1. 29.9750^\circ
  2. 31.1375^\circ
  3. 76.3467^\circ

Exercise Answers

  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. ↵

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Technical Mathematics, 2nd Edition Copyright © 2024 by Morgan Chase is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.