Transcript The Extraordinary Sums of Leonhard Euler

Chapter 1: Tools of Geometry
Lesson 1-4: Measuring Segments
and Angles
Goal: Find the lengths of segments
and the measures of angles.
Quote for today:
“Why are numbers beautiful? It's like
asking why is Beethoven's Ninth
Symphony beautiful. If you don't
see why, someone can't tell you. I
know numbers are beautiful. If they
aren't beautiful, nothing is.”
-Paul Erdös
Finding Segment Lengths:
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Postulate 1-5 Ruler Postulate:
The points of a line can be put into
one-to-one correspondence with the
real numbers so that the distance
between any two points is the
absolute value of the difference of
the corresponding numbers.
Finding Segment Lengths:
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The length of AB is
given by AB = |a – b|,
where a is the coordinate of A
and b is the coordinate of B.
Two segments with the same
length are congruent segments.
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If AB = CD, then AB ≅ CD.
Finding Segment Lengths:
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Postulate 1-6 Segment Addition
Postulate: If three points A, B, and C
are collinear and B is between A and
C, then AB + BC = AC.
A midpoint of a segment is a point
that divides a segment into two
congruent segments.
A midpoint, or any line, ray or other
segment through a midpoint, bisects
the segment.
Finding Angle Measures:
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An angle is formed by two rays with
the same endpoint.
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The rays are the sides of the angle.
The endpoint is the vertex of the angle.
Finding Angle Measures:
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Postulate 1-7 Protractor
Postulate: Let OA and OB be
opposite rays in a plane. OA, OB, and
all the rays with endpoint O that can
be drawn on one side of AB can be
paired with the real numbers from 0°
to 180° so that
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OA is paired with 0° and OB is paired with
180°.
If OC is paired with x° and OD is paired
with y°, then m∠COD = |x° – y°|.
Finding Angle Measures:
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We can classify angles according to
their measures.
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acute angle
0° < x < 90°
right angle
x = 90°
obtuse angle
90° < x < 180°
straight angle
x = 180°
Finding Angle Measures:
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Postulate 1-8 Angle Addition
Postulate:
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If point B is in the interior of ∠AOC,
then m∠AOB + m∠BOC = m∠AOC.
If ∠AOC is a straight angle,
then m∠AOB + m∠BOC = 180°.
Angles with the same measure are
congruent angles.

If m∠1 = m∠2, then ∠1 ≅ ∠2.
Assignments and Note:
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CW: Enrichment 1-4.
HW 1-4: #2-28 (evens), 60-70
(evens); 79 is extra credit.
Get geometry tools: compass, ruler,
protractor, and straightedge.
Test next Wednesday.