Measuring Angles

Today’s Learning Goals We will learn that measuring angles is nothing but COUNTING the number of units that fit into the angle you are measuring.

Measurement We measure all different types of things in this world. We often measure the length of something for example. If I wanted to measure the length of the table in the front of the room, what would I do? Most people think of using a ruler to measure length and COUNTING the number of inches or feet. Could we measure the length of the table with a bunch of paper clips? If so, explain how and if not, explain why. Yes. You put the paper clips end to end and COUNT how many paper clips fit in the length of the table.

Measurement In Ancient Times, the King’s foot was used as a unit of measurement for length. Everyone in the land would get a piece of wood that was the length of the King’s foot and COUNT how many of the King’s feet fit into the thing they were measuring the length of. However, when the King changed, everyone had to change their piece of wood to measure the length of something. After a while, this got old, and someone said, “Let’s come up with a standard unit of measurement that everyone will use to measure length. Then we won’t have to change units when we get a new King!!”

Measurement So, now we have a standard “foot” unit and other standard units to COUNT or measure the length of something. Notice, that to measure the length of something, you use a smaller unit of length to measure it!

Measurement We measure things other than length by COUNTING the number of units that fit into it. What are some other things that we measure other than length? Weight – COUNT how many smaller units of weight fit Area – COUNT how many smaller units of area fit Volume – COUNT how many smaller units of volume fit Temperature – COUNT how many smaller units of temperature fit

Angles An angle is the union of two rays or line segments. Angles are another thing that are measured. To measure angles, we COUNT how many smaller angles fit into the angle we’re measuring. The point where the two rays or sides meet is called the vertex. vertex

Measuring Angles Let’s make a non-standard unit of measurement to measure the first angle on your worksheet. Let’s call this unit an octon. Take an 8½” x 11” piece of paper and fold it as shown in the illustration at the left.

vertex Non-standard Unit for Measuring Angles Yes…put one octon next to each other and COUNT the number of octons that fit in. According to the picture above, about how many octons fit into the angle? Nice…about 6½ octons. If we let the “turn” or small wedge in our octon below be our measuring unit, how could we measure the angle at the right? ½

Measuring Angles Notice that we estimated about 6½ octons fit into the angle. How could we get a better estimate of the measurement of the angle? Right…get a smaller unit of measure. A standard unit of measure that is used to measure angles is called the degree. An angle of 1 degree (also written 1  ) is a very small turn, or a very narrow wedge. 11

Measuring Angles The size of a degree was chosen so that a right angle has a measure of 90 . Here is a 90  angle. Imagine 90 copies of the 1  angle fitting into this angle. 90 

Measuring Angles If 90  is in a right angle, how many degrees would be in a straight angle (shown below)? 90  Yes…180 .

90  Estimating Measurements of Angles You can use the 90  (right angle) and the 180  (straight angle) to help you make estimates on the angle measurements of other angles. For example, is the angle below more or less than 90  ? Great…this angle is less than 90 . Angles less than 90  are called acute angles. To help people remember the word acute, some people think little angles are “cute”.

90  Estimating Measurements of Angles Is the angle below more or less than 90  ? Great…this angle is more than 90 . Angles more than 90  are called obtuse angles. To help remember the word obtuse, some people think of “obese” (big) angles.

Measuring Angles To determine close estimates to the measure of an angle, a protractor is often used. A protractor is a tool similar to a ruler. It has degree units already marked on it.

Measuring Angles To get a better estimate to the exact measurement, we could use a protractor. Previously, we estimated that the angle below was less than 90 . Vertex in hole One ray on 0  – 180  line. Place the small hole of the protractor on the vertex of the angle and one ray on the 0  – 180  line.

Measuring Angles Some students might say the measurement is around 110  and other students might say it is around 70 . Which one do you think is correct? Explain why. Nice…this angle is around 70 . It can’t be 110  because we know from our estimate that it is less than 90  !

Naming Angles Usually, angles will have points with letters placed on them so that you can give names to the angles. The vertex point is usually one point that is given. B We could name this angle  B. The symbol “  ” is read “angle”.

Naming Angles Now consider the following diagram. Do you think it is still okay to name it  B? B No…because we would not be sure which angle we were talking about. Suppose we put more points on the figure. What angles do you see? C A D Great…  ABC,  ABD,  DBC. Notice that the vertex is always the middle letter when naming an angle.

Naming Angles Another way to name angles is with numbers. B C A D 1 2  1 is the same as what angle name? Good…  1 =  ABC.  2 is the same as what angle name?  1 +  2 is the same as what angle name? Nice…  2 =  DBA. Excellent…  1 +  2 =  DBC.

Measuring Angles Consider the angle below. Is it more or less than 90  ? Can we use the protractor the way it is placed above to say that the angle is around 135  ? 1 Great…it is more than 90 . No…because the ray of the angle is not on the 0   line.

1 Measuring Angles How do we have to align the protractor to determine the measure of  1? Excellent…put the vertex at the hole in the protractor and line up one ray with the 0  – 180  line. Nice…m  1 is around 146 . What is the m  1 (read “measure of angle 1”)? 1

Partner Work You have 15 minutes to work on the following problems with your partner.

For those that finish early Look around this classroom. Find at least one example of each type of angle described: a) An angle that occurs as the result of a turning motion, such as the opening of a door. b) An angle that occurs as a wedge, such as a piece of pizza. c) An angle that occurs as two sides with a common vertex, such as the branches on a tree.

Big Idea from Today’s Lesson We can use 90  and 180  as benchmarks to help us estimate the measurement of an angle. Measuring an angle is similar to all other types of measurement…to measure, we COUNT the number of smaller units of an angle that fit in. A degree is the standard unit for measuring angles.

Homework Pgs. 215 – 216 (2 – 20) You need a protractor to answer some of the problems.