Math 10-3

Unit 5 Angles and Parallel Lines

Lesson 51 Measuring, Drawing, and Estimating Angles

Formula page

I expect that you will be able to use a formula page for this class,  but that will be determined by your teacher.

Your teacher may have a formula page they expect you to use.

Below is a link to the formula page I will use (at least for now!) when I’m doing questions for this class.

This may change, depending on feedback from your teacher.

I encourage you to download this page and print it so you have it handy when you are doing math.

LINK
https://drive.google.com/open?id=1EMauePx-P6WrtTIcd7-AtNnmJt8zqXpI

Geometry Constructions

This is a “pure math” topic, that explores constructing various things (angles, certain triangle, etc.) using JUST a simple straight edge (with no markings)  and a compass (used to make circles)

Your teacher may or may not want you to do this.

I expect most students in this class will be interested in more practical methods to do the same thing, and that is what I will focus on.

Students in this class should have a geometry set with a compass, ruler, protractor and a couple of triangles (45 - 45 - 90 and 30 - 60 - 90).

Those tools will enable you to do any of the required constructions, much more easily in some cases.

This will be my focus in these lessons, but as I mentioned, you will need to ask your teacher directly if you will be expected to know the “pure math” constructions as well.

Explore the Math Pages 174 - 175

Read/Work through this section of your textbook.
These ideas will be developed in the examples

Your teacher will let you know which pages you need to do for this lesson.

Below,  some extra information to help illustrate the ideas of this lesson.

Most of this is review,  and a reminder of important vocabulary.

We need to have this shared understand of words and ideas to be able to talk about the ideas in these lessons.

What follows here just gives MORE information than the textbook, in case you need it to understand.

What is an angle?    Demonstration

💡  Definition:	An angle is formed by two rays meeting at a common vertex

Video Tutorial  (Angle A)

Review  classifying angles

Video Tutorial  (Angle B)

Math Open Reference
https://www.mathopenref.com/angleacute.html

Angles are:

• acute, if their measure is between 0° and 90°;

• right, if their measure is 90°; the two rays are perpendicular to each other;

• obtuse, if their measure is between 90° and 180°;

• straight, if their measure is 180°; and

• reflex, if their measure is between 180° and 360°.

Angle measure (a common error)

Some students think that the blue angle above is bigger,  since the space between the arrowheads is much larger than the space between the red arrowheads.

An understandable error!

However,  both angles shown above have a measure of 30°,  and it's very important to recognize that they are equal.

The definition of the measure of an angle being a measure of the opening between the rays isn't very helpful since the blue arrow heads ARE farther apart!

So,  it can be helpful to think of angles in another way.

If you think of an angle as being a certain chunk of a circle,  that can work.

Circles can have different sizes,  but both the above angles can be thought of as taking the same relative sized piece  out of each circle,  as the following demonstrations will show.

The length of the rays that indicate an angle don't have anything to do with the actual measure of the angle.

Think of an angle as being a certain portion of a circle.

In the diagram below,  both angles shown have a measure of 40°,  even though the blue angle looks larger .

In each case, the relative size of each piece is the same, it takes the same relative sized piece of pie from each circle.  
(so the angle is the same,  but yes... the piece of pie from the blue circle is a bigger PIECE!  but the angle formed by the piece of pie in each case is the same)

Demonstrations  Angle “size”

Video Tutorial  (Angle C)

Angle Measure in degrees

Video Tutorial  (Angle D)

Experiment with the demonstration below.

You can control the number of equal parts you divide the circle into.

You can control the position of the angle.

You can control whether to display the measure of the angle or not.

Try several examples and determine what the angle is and then reveal the measure to see if you are right.

Dividing up a circle

💡  Definition:	An angle is the measure of the spread between two rays.

💡  Definition:    One degree is the angle subtended (marked off)  by one 360th part of the circumference of a circle.

We divide a circle up into 360 parts.

ONE DEGREE is the spread between two points on a circle whose circumference has been divided into 360 equal sections.

Clock face angles  (referents)

Video Tutorial  (Angle E)

Supplementary and Complementary angles

Video Tutorial  (Angle F)

Complementary angles: two angles that have measures that add up to90°
Complementary angles    LINK

Supplementary angles: two angles that have measures that add up to 180°
Supplementary angles    LINK

True Bearing

Video Tutorial  (Angle G)

In navigation and map-making, people often measure angles from the vertical, or north.

The angle, measured in a clockwise direction from a line pointing north, is referred to as the true bearing.

Straight north has a bearing of 0°.

true bearing: the angle measured clockwise between true north and an intended path or direction, expressed in degrees

Example 1 Pages 175 - 178

Read the example problem and try to answer the question YOURSELF if you can.
Use the textbook solution or the Video Tutorial if you need help.

Video Tutorial

IF you are interested (and maybe your teacher will REQUIRE you to do questions this way),   the Math Open Reference website has LOTS of examples of the “pure math” geometry construction techniques.

Check here
https://www.mathopenref.com/constructions.html

Specifically for EXAMPLE 1,  here is the page that illustrates how to copy an angle using ONLY  a compass and a straight edge.
https://www.mathopenref.com/constcopyangle.html

Example 2  Page 180

Read the example problem and try to answer the question YOURSELF if you can.
Use the textbook solution or the Video Tutorial if you need help.

Video Tutorial

Example 3  Page 180

Read the example problem and try to answer the question YOURSELF if you can.
Use the textbook solution or the Video Tutorial if you need help.

Video Tutorial

Example 4  Page 181

Read the example problem and try to answer the question YOURSELF if you can.
Use the textbook solution or the Video Tutorial if you need help.

Video Tutorial

Example 5  Page 180

Read the example problem and try to answer the question YOURSELF if you can.
Use the textbook solution or the Video Tutorial if you need help.

Video Tutorial

Mental Math and Estimation   Page 179

Do your best to use mathematics in your head to solve this problem.
Use the Video Tutorial to check your answer or get help with the problem.

Video Tutorial

Build Your Skills Pages 184 - 185

These questions all come from your textbook.

If you need extra practice or want to maintain your skills,  see the NEXT section for more practice questions.

Doing Practice Problems

The problems in this section reinforce the math presented in this lesson, and give you opportunities to build on and practice your skills.

•	Work through the practice problems for this lesson.
	These problems are for practice, you don’t need to hand them in!
	(your teacher may assign specific questions for you to complete)

•	Use the answers in the back of the book to CHECK YOUR WORK!

•	If you make a mistake, TRY TO FIX IT!

•	If you need help or want more information, view the video solutions provided.

•	If you still need more help,  CONTACT YOUR TEACHER!

•	EXTEND YOUR THINKING problems will be more challenging than the others.

Notes

For all these questions, the answers in the back of the book are (hopefully) straight-forward,  and should give you the feedback you need.
These questions are “one step” and there really isn’t any value added for me to do them in video form.
If you want more help and feedback,  check the extra practice problems provided.
I made a video for all of those.

Question 1
The answers in the back of the book give you the exact measures, as well as some good estimates

Question 6
Would be great to make a map like this,  but that might be something your teacher assigns as more of a “project” kind of thing.

Question 7
A great question, but pretty complicated!
The answer in the back of the book is helpful and shows what to do.
It can get complicated.
If you are interested,  do a search for magnetic north movement.... it’s always moving!
So declination is changing all the time....

Develop and Maintain Your Skills

Doing these problems

These questions are provided to help you develop and maintain your skills with the ideas of this lesson.

These questions are modelled after the questions in the textbook,  so there is nothing different here, just more questions if you need them.

These questions are completely optional (unless your teacher specifically assigns them)

Each question has an answer AND a video solution.

DO your best to answer the question YOURSELF first!

Check your work with the answer provided.

If you make a mistake,  try to fix it!

Use the video solutions for more information.

If a problem still doesn't make sense, ASK YOUR TEACHER FOR HELP!

Practice Problems

Problem

Answer

a) This is a right angle.
b) This is a straight angle.
c) This is an obtuse angle.
d) This is an acute angle.
e) This is a reflex angle.

Video Solution Link

Problem

Identify the type of angle: 
acute, right, obtuse, straight, or reflex. 
a) 68° 	b) 215° 	c) 91° 	d) 32°

e) 180°	f) 99°	g) 195°	h) 265°

Answer

a)	acute
b)	reflex
c)	obtuse
d)	acute
e)	straight
f)	obtuse
g)	reflex
h)	reflex

Video Solution Link

Problem

Given each of the following angles, determine the size of the complement and/or the size of the supplement 
if they exist).

a) 75°	b) 43°	c) 103°	d) 87°	e) 300°

Answer

Video Solution Link

Problem

Fill in the chart with the complement and the supplement of each angle, if they exist. 
If they don’t exist, state why.

Answer

Video Solution Link

Problem

The complement of an angle is 58°.

a) What is the size of the angle?

b) What is the supplement of the angle?

Answer

a)	32°
b)	148°

Video Solution Link

Problem

A boat is heading directly southwest. 
What is its true bearing?

Answer

If the boat is heading southwest, measuring from the vertical will give you an obtuse angle of 225° (45° beyond a straight angle).

Video Solution Link

Problem

If a boat is travelling 25° south of straight east, what is its true bearing?

Answer

The true bearing is 115°

Video Solution Link

Problem

What is the true bearing of a boat travelling south?

Answer

180°

Video Solution Link

Problem

If Renata cuts a rectangular tile diagonally, one of the acute angles formed is 65°.

What is the size of the other acute angle?

Answer

25°

Video Solution Link

Problem

Pete is laying irregularly shaped paving stones. He needs to find one to fit in position A. 
Approximately what size of angle will it have at the corner shown?

Answer

The angle has a measure of 110°

Video Solution Link

Problem

What time is indicated on each clock?
What angle has the minute hand moved from the first time to the second time?

Answer

The time on the first clock is 4:00
The time on the second clock is 4:23
The minute hand has moved through 138°

Video Solution Link