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Saxon Math 8/7 Homeschool: Online Textbook Help34 chapters | 191 lessons

Instructor:
*Michael Quist*

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

In real life it takes special equipment to accurately measure angles, but you can make a pretty good guess if you know a few tricks. We'll discuss some of those in this lesson.

You've created the perfect ramp for your skateboard jump. It gives you just that amount of lift you need, your tricks are working perfectly, and your landings look like they belong in the Olympics. Then the community planning committee announces that your ramp is too steep; it has to be less than 30°. After you get over the desire to throw a fit, you stop and think, ''Wait a minute! Are they right? Is my ramp really steeper than 30°?''

**Estimating an angle** means trying to make an educated guess about an angle measurement. In this lesson, we'll explore some tricks that make this easy.

When you're estimating an angle, first decide which quadrant it would land in, if you put one side on the 'starting point', the 0° line between Q1 and Q4, and then let it stretch counterclockwise around the quadrants. Some angle types are listed below:

Angle Type | Measure Range | Lands in Quadrant |
---|---|---|

acute | 0° to 90° | Q1 |

right | 90° | between Q1 & Q2 |

obtuse | 90° to 180° | Q2 |

straight | 180° | between Q2 & Q3 |

reflex | 180° to 360° | anywhere in Q3 or Q4 |

First, let's eliminate the easy ones. If your angle lines up perfectly with the 0° line and one of the other lines between the quadrants, then you don't need to estimate.

- Right (90°) angle
- Straight line (180° angle)
- 3/4 of a circle (270° angle)

Assuming it's not one of those, then it's time to go to work. If it's not **acute** (between 0° and 90°), then mentally cut out as many 90° blocks as you can, (setting them aside to add in later) until you are left with an angle that is less than 90°.

Now, mentally rotate your remaining angle so it lines up with the picture above (the area of the angle should span across Q1, with one side of it resting on the 0° 'starting' line). 45° is the center between 0° and 90°. Is your remaining angle larger or smaller than that? Is it a 60°? A 30°? Perhaps a 75° or a 15°? If it doesn't really match any of those, then try and estimate within 5°:

- larger than 75°: use 80° or 85° (depending on which seems closer)
- between 60° and 75°, use 65° or 70°
- between 45° and 60°, use 50° or 55°
- between 30° and 45°, use 35° or 40°
- between 15° and 30°, use 20° or 25°
- between 0° and 15°, use 5° or 10°

Once you've picked the closest acute angle, make sure you add 90° for each 90° block you removed (if any). For example, say you removed two 90° blocks from your angle and had 25° left. Your final angle would be:

25° + 90° + 90° = 205°.

Since our skateboard ramp example isn't an angle on paper, we'll use the **tangent** (the ratio between height and length) to get us close.

Tangent/Slope Ratio | Approx. Angle |
---|---|

1/5 | 11° |

1/3 | 18° |

1/2 | 27° |

2/3 | 34° |

1 | 45° |

2 | 63° |

3 | 72° |

4 | 76° |

5 | 79° |

10 | 84° |

20 | 87° |

Now obviously you probably don't want to memorize all of these, but it helps to at least know that a tangent of 1 is 45°, a 1/2 is 27°, and a 3 is 72°. Okay, so here's the plan for estimating an acute angle outside (if it's not acute, mentally chop off 90° blocks again, until you get down to acute):

- Take a stick, your arm, or whatever, and measure the height and the length.
- Divide the height by the length. That will give you the tangent.
- Remember what you can of the tangent/angle table. Using your tangent ratio, you should be able to get a pretty good idea of what the angle is.

Back to the ramp example. Say you measure the length, and it's five stick lengths, while the height is only one stick length. That would be a tangent of 1/5. That's only an 11° angle! No way your ramp is anywhere near the 30° they said it was! In fact, if it's anything less than a 3/5 ratio of height to length, which is likely, you're probably fine.

We can guess that a 90° will happen at about the blue line. Looking at the remaining angle, we can tell that it's definitely less than 45°, maybe about a 30° angle. Add that to the 90° we already marked off, and we have about a 120° angle.

This is already an acute angle. Larger than 45°, larger than 60°, maybe a 75°?

Okay, how about this drooling PacMan character? You can see that the angle is bigger than 270°, so you'll want to chop it off at the pink line, then see what's left. Looks like about 45°. Add that to the 270° that we chopped off, and we have about a 315° angle, give or take. Notice, we could also subtract our 45° from 360° (whole circle), to get to the same place.

**Estimating an angle** means trying to guess at the measurement of the angle. If it's larger than 90°, chop off as many 90° sections as you can (and put them aside to add in later), until you have the **acute** (between 0° and 90°) part. You can get a good estimate of that acute angle by comparing it to known angles, such as 30°, 45°, and 60°. You can also use the **tangent** (height/length) to estimate it. Estimating angles isn't too hard to do, and it can save you a lot of grief when the neighborhood committee comes to criticize your skateboard ramp!

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Saxon Math 8/7 Homeschool: Online Textbook Help34 chapters | 191 lessons

- Interior and Exterior Angles of Triangles: Definition & Examples 5:25
- Types of Angles: Vertical, Corresponding, Alternate Interior & Others 10:28
- Complementary Angles: Definition, Theorem & Examples 4:24
- Supplementary Angle: Definition & Theorem 4:29
- Angles Formed by a Transversal 7:40
- How to Measure Angles with a Protractor 3:02
- How to Find the Number of Diagonals in a Polygon 4:49
- How to Measure the Angles of a Polygon & Find the Sum 6:00
- How to Estimate Angle Measures
- Go to Saxon Math 8/7 Homeschool: Angles

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