Download §3.2 Corresponding Parts of Congruent Triangles

§4.1 Triangles
The student will learn about:
medians,
altitudes, and
other geometric
properties of
triangles.
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§4.1 Congruent Triangles
We will now study some of the special
segments, lines and rays associated with
a triangle.
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Median
Definitions
The median of a triangle is a segment
whose end points are a vertex of the
triangle and the midpoint of the opposite
side.
3
Theorem
The medians of a triangle concur.
Proof: For this proof we will need
parallelograms later in the course.
4
Angle Bisector
Definitions
A segment is an angle bisector of a triangle
if it lies on the ray which bisects an angle
of the triangle and its endpoints are the
vertex of this angle and a point on the
opposite side.
5
Theorem
The angle bisectors of a triangle concur.
Proof for homework.
Strategy!
in pairs
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Perpendicular Bisector
Definitions
A line is a perpendicular bisector of a side
of a triangle if it is perpendicular to that
side and passes through the midpoint of
that side.
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Theorem
The perpendicular bisector of a segment, is
the set of all points of the plane that are
equidistant from the endpoints of the
segment.
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Proof.
B
D
Given:
with midpoint C
What isAB
given?
and CD perpendicular to
the segment.
(1) AC = CB.
(2) Construct AD and BD
(3) BCD = ACD
(4) CD = CD
(5) ∆ACD = ∆BCD
(6) AD = BD
C
A
Prove:will
What
ADwe
= BD
prove?
Why?
Given
Construction
Why?
Right angles
Why?
Reflexive
Why?
SAS
Why?
CPCTE
Why?
QED
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Theorem
The perpendicular bisectors of a triangle
concur.
Proof left as a homework assignment. Hint:
two must concur, show that the third
must concur at the same point.
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Altitude
Definitions
The altitude of a triangle is a perpendicular
segment from a vertex of the triangle to
the line containing the opposite side.
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Theorem
The altitudes of a triangle concur.
Proof left as a homework assignment. Hint:
Q
U
C
T
P
O
A
B
V
R
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Exterior Angle
Definitions
If A – C – D, then BCD
is an exterior angle of
∆ ABC.
B
D
A
C
Every triangle has six exterior angles. These
form three pairs of vertical angles.
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Theorem
The exterior angle of a triangle is greater
than each of its remote interior angles.
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Proof
Given: A – C – D and ∆ ABC
Prove:  BCD >  B
Prove  ACG >  A (homework)
B
F
E
A
C
D
G
Statement
Reason
Do
in class as a ”cut and paste proof!”
1. Let E be the midpoint of BC.
2. Choose F on AE so that AE = EF
5. m  B = m  ECF
Construction
Construction
Vertical angles
SAS
CPCTE
6. mB + m FCD = mECF + m FCD
Arithmetic
3.  BEA =  CEF
4. ∆AEB = ∆FEC
continued
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B
F
E
A
C
D
G
6. mB + m FCD = mECF + m FCD
Arithmetic
7. mBCD = mECF + m FCD
Angle Addition
8. mBCD = mB + m FCD
CPCTE & Substitution
9. mBCD > mB
Arithmetic
10. BCD > B
Angle measure
postulate
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Before we continue our study of
quadrilaterals we will need the
following information on parallelism.
Historical Background
Euclid’s Fifth. If a straight line falling on two
straight lines makes the interior angles on the
same side less than two right angles, the two
straight lines, if produced indefinitely, meet on
that side which are the angles less than the two
right angles.
n
l
A
m
 1 +  2 < 180 then
lines l and m meet on
the A side of the
transversal n.
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Playfair’s Postulate
Given a line l and a point P not on l, there exist
one and only one line m through P parallel to l.
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Equivalent Forms of the Fifth
 Area of a right triangle can be infinitely large.
 Angle sum of a triangle is 180.
 Rectangles exist.
 A circle can pass through three points.
 Parallel lines are equidistant.
 Given an interior point of an angle, a line can be
drawn through the point intersecting both sides of
the angle.
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Euclidean Parallelism
Definition. Two distinct lines l and m are said
to be parallel, l || m, iff they lie in the same
plane and do not meet.
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Theorem 1: Parallelism in
Absolute Geometry
If two lines in the same plane are cut by a
transversal so that a pair of alternate interior
angles are congruent, the lines are parallel.
Notice that this is a theorem and not an axiom
or postulate.
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Parallelism in Absolute Geometry
Given: l, m and transversal t. and
1≅ 2
Proof by contradiction.
(1) l not parallel to m, meet at R.
Prove: l ‫ ׀׀‬m
(2)  1 is exterior angle
Assumption
Def
(3) m  1 > m  2
Exterior angle inequality
(4) → ←
Given  1 ≅  2
l
t
A
1
m
2
B
R
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Parallelism in Absolute Geometry
Given: l, m and transversal t. and
1≅ 2
Proof by contradiction.
(1) l not parallel to m, meet at R.
Prove: l ‫ ׀׀‬m
(2)  1 is exterior angle
Assumption
Def
(3) m  1 > m  2
Exterior angle inequality
(4) → ←
Given  1 ≅  2
l
t
A
1
m
2
B
R
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Assignment: §4.1
You will need a straight edge
and compass for the next two
lessons. Buy them.