MA.8.G.2.2
Classify and determine the
measure of angles, including
angles created when parallel
lines are cut by transversals.
Block 25
Some History
The history of the
mathematical
measurement of angles,
possibly dates back to
1500BC in Egypt, where
measurements were
taken of the Sun's
shadow against
graduations marked on
stone table.
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Why 360° in the Circle?
• Probably based on the fact that 360 is
approximately the number of days in a
year.
• Its use is often said to originate from
the ancient Babylonians (1696 – 1654
BC). Ancient astronomers noticed that
the stars in the sky seem to advance in
a circular motion by approximately
one-360th of a circle, i.e., one degree,
each day.
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Subdivisions of Degrees
• One degree is divided into 60 minutes
(of arc)
• One minute is divided into 60 seconds
(of arc)
• Another indication of the Babylonian
influence
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Example on Earth
• Longitude is the angular distance, in
degrees, minutes, and seconds, of a
point east or west of the Prime
(Greenwich) Meridian.
• If you divide the circumference of the
earth (approximately 25,000 miles) by 360
degrees, the distance on the earth's
surface for each one degree longitude is
just over 69 miles, or 111 km.
• There are 24 standard time zones defined
by using the meridians. How many
degrees between each time zone?
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Divisors of 360
• 360 has 24 divisors, including 1 and 360
• Can you find all the divisors?
• The divisors include every number from
1 to 10 except 7
• For the number of degrees in a circle
to be divisible by every number from 1
to 10, how many degrees would the
circle need?
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Vocabulary
In everyday language:
• Obtuse means not alert in perception or
intellect
• Acute means sharp
• Supplementary means additional
• Complementary means completing
Do the everyday meaning of these words
relate to their mathematical meaning?
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Vertical - Adjacent Angles
Open the GeoGebra file vertical_adjacent
1. Name some facts about the observed angles in
figures 1, 2, and 3
2. What are congruent angles?
3. What figure shows vertical angles?
4. What figure shows adjacent angles?
5. Is it enough to define vertical angles as opposite
angles?
6. Is it enough to define adjacent angles as next to
each other?
7. What would be an accurate definition for
vertical angles and adjacent angles?
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Students Confuse…
• Adjacent angles
• Linear pair
• Supplementary angles
– Draw examples of each one or create a
GeoGebra file that shows clearly the
difference between them.
– How can we help students understand
and remember the difference between
them?
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Corresponding Angles
• Open the GeoGebra file corresponding1
• Move point A and notice the different
pairs of corresponding angles
• Overlap point A with point B and move
the point on the slider to the left. What do
you notice?
• How can this file help teach this concept
to the students?
• Could you make a similar file to show the
definition for Alternate Interior angles?
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Corresponding Angles
• Open the GeoGebra file
corresponding2
• Move points A or B:
– What do you notice?
– What is the relationship between the blue
lines when the pairs of corresponding
angles have the same measure?
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Interactive Website
http://www.mathopenref.com/anglesco
rresponding.html
Explore this website and its links to other
angle relationships.
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Angles in a Clock
Open the GeoGebra file angles_clock
1. How many degrees separate one number
from the next?
2. How many minutes must pass for the hour
hand to move 1 degree?
3. What is the angle measure between the hour
hand and the minute hand at 12:15?
4. When after 3 o’clock will the angle between
the hour and minute hands be exactly 180°?
You can use a protractor to verify your answers
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