Download 10.1 Lines, Angles, Circles

10.1 Lines, Angles, Circles
Classical Geometry is the study of points,
lines, angles, circles, etc and the
geometric figures built out of them.
The ideas and definitions we use today go
back to the mathematician Euclid from
Alexandria in 300BC in his book Euclid's
Elements.
A point is “that which has no part”.
A line has “length but no breadth.”
A plane has “length and breadth only.”
Points, Lines, and Planes
Euclid: a point is “that which has
no part,” a line has “length but no
breadth,” and a plane has “length
and breadth only.”
A point on a line divides the line
into three parts—the point and
two half lines. A ray is a half line
with its endpoint included.
A piece of a line joining two
points and including the points is
called a line segment.
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Section 10.1, Slide 6
Points, Lines, and Planes
Parallel lines lie on the same
plane and have no points in
common.
Intersecting lines lie on the
same plane and have a single
point in common.
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Section 10.1, Slide 7
Pop Quiz!!!
The type of object is
1) A ray
2) A line
3) A line segment
4) None of the above
Angles
Two rays having a common endpoint form an angle.
We measure angles in units called degrees. The
symbol ° represents the word degrees.
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Section 10.1, Slide 8
Let's start with an angle where the initial
side and terminal side are the same (i.e.
go all the way around). What is the
measure of that angle?
Angles
An angle whose measure is
between 0° and 90° is called
an acute angle.
A right angle has a measure
of 90°.
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Section 10.1, Slide 10
Angles
An obtuse angle has a measure
between 90° and 180°.
A straight angle has a
measure of 180°.
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Section 10.1, Slide 11
Angles
Two intersecting lines form two pairs of angles
called vertical angles.
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Section 10.1, Slide 12
Angles
A pair of angles is complementary
if the sum of their measures is 90°.
Two angles having an angle sum
of 180° are supplementary angles.
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Section 10.1, Slide 13
Let's see how to prove this.
Pop Quiz!!!
The type of angle is
1)
2)
3)
4)
5)
6)
7)
8)
Acute
Obtuse
Right
Straight
Vertical
Complementary
Supplementary
None of the above
Angles
Two lines that intersect forming right angles are
called perpendicular lines.
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Section 10.1, Slide 17
Angles
If we intersect a pair of parallel lines with a third
line, called a transversal, we form eight angles.
●
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Section 10.1, Slide 18
Angles
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Section 10.1, Slide 16
Angles
• Example: If lines l and m
are parallel, find the measure
of the other angles.
Angles
• Example: If lines l and m
are parallel, find the measure
of angle 9.
• Solution:
(corresponding angles)
(straight angle)
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Section 10.1, Slide 18
Angles
• Example: If lines l and m
are parallel, find the measure
of angle 2.
• Solution:
(same side interior angles)
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Section 10.1, Slide 19
Circles
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Section 10.1, Slide 20
Circles
An angle that has its vertex
at the center of a circle is
called a central angle.
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Section 10.1, Slide 21
Circles
• Example: A circle has a circumference of 12
meters. If central angle ACB has measure of
120°, then what is the length of the arc from A to
B?
(continued on next slide)
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Section 10.1, Slide 22
Circles
• Solution:
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Section 10.1, Slide 23
Pop Quiz!!!
The type of object is
1) The center
2) The diameter
3) The radius
4) A central angle
5) None of the above
The same Eratosthenes that found the
prime number sieve also was the first
person to prove that the earth was round.
He accurately determined the
circumference of the earth.
Circles
• Example: Use elementary geometry to
estimate the circumference of Earth.
• Solution: Assume that lines l and m are parallel
and cut by the transversal t. The point C is the
center of the circle. Therefore, angles α and β
are equal.
(continued on next slide)
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Section 10.1, Slide 24
Circles
To measure the circumference of Earth, place a
vertical pole in the ground and wait until noon
when the rays of the Sun and the pole form an
angle of 0°.
Suppose at that very moment, a friend 1,000
miles away also has a similar vertical pole, and
the Sun’s rays make an angle of 15° with his
pole.
(continued on next slide)
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Section 10.1, Slide 25
Circles
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Section 10.1, Slide 26