Congruent Segments

1
Congruent Segments
When two segments have the same length, they are
said to be congruent segments.
Measure of segments
Congruent
B
C
If AB AC
A
Is read Segment AB is congruent to segment AC
then AB AC
Segments
Is read The measure of segment AB is equal to
the measure of segment AC
Do modeling activity on page 36 Locating the
midpoint of a segment
You can also compare the measure of segments. For
example you can say AB lt BC or BC gt AC
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MIDPOINTS AND SEGMENT CONGRUENCE
The midpoint of a segment is the point
equidistant from the endpoints of the segment.
.
.
.
Q
P
M
Definition of Midpoint The midpoint M of PQ is
the point between P and Q such that PMMQ
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Midpoint Formulas

• On the number line, the coordinates of the
  midpoint of a segment whose endpoints have
  coordinates a and b is ab
• 2
• In a coordinate plane, the coordinates of the
  midpoint of a segment whose endpoints have
  coordinates (x1, y1) and (x2, y2) are
• (x1x2, y1y2 )
• 2 2

4
Examples
Example 1 (p.37)
a) Use the number line to find the coordinates of
the midpoint of FG
G
0
F
-3 2 1 0 1 2 3 4 5 6 7 8
b) Find the coordinates of Q, the midpoint of
RS, if the endpoints of RS are R (-3,-4) and S
(5,7).
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More examples

• Example 2 p. 38 (Students do then I explain)
• Find the coordinates of point Q if L(4,-6) is
  the midpoint of NQ and the coordinates of N are
  (8,-9)
• If Y is the midpoint of XZ, XY 2a 11, and YZ
  4a 5, find the value of a and the the measure
  of XZ

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Segment Bisector
Segment bisector is a segment line or plane that
intersects a segment at its midpoint.
.
R
.
.
.
.
.
Q
P
M
M
N
.
M, TM, RM and plane N are all bisectors of PQ
T
Do construction p. 39
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The Midpoint Theorem
(Theorem 1-1) Midpoint theorem If M is the
midpoint of AB then AM MB

• Proof of theorem 1-1 Given that M is the
  midpoint of AB, write a paragraph proof to show
  that AM MB.

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What are proofs?
A proof is a logical argument in which each
statement you make is backed up by a statement
that is accepted as true.
Paragraph proof or informal proof a paragraph
that explains why a conjecture for a given
situation is true. Conjecture is an educated
guess.
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Paragraph proof of theorem 1-1
From the definition of midpoint of a segment, we
know that AM MB.

That means that AM and MB have the same measures
By the definition of congruence, if AM and MB
have the same measure, they are congruent
segments.
A.
M.
Thus, AM MB.
B .