Paul Hein

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Chapter 9
Paul Hein Period 2 12/12/2003
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(No Transcript)
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Parallelism
Key Terms
Skew Lines Skew lines are 2 lines that are
neither parallel nor intersect thus, they must
be in different planes.
L1 and L2 are intersecting lines. L1 and L3 are
parallel lines. L2 and L3 are Skew lines.
Transversal A transversal is a line that
intersects two coplanar lines.
L1 and L2 are coplanar. Thus, line T is the
transversal.
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Alternate Interior Angles Alternate interior
angles are formed when 2 lines are cut by a
transversal. They are any two angles that Are on
opposite sides of the transversal, are formed by
2 different coplanar lines, and are on the
interior of the parallel lines.
L1 and L2 are 2 coplanar lines. T is a
transversal of them. ?A is an alternate interior
angle to ?C, and ?B is an alternate interior
angle to ?D.
Interior Angles on the same side of the
transversal These are exactly what they sound
like IF you have 2 coplanar lines that are Cut
by a transversal, then any two angles on the
interior of the parallel lines and on the same
side of the transversal fit this Description.
(fig above) L1 and L2 are coplanar lines. T is a
transversal of them. ?A is an interior Angle on
the same side of the transversal to ?D, and ?B is
an interior angle on the same side of the
transversal with ?C.
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Corresponding Angles If you have two coplanar
lines cut be a transversal, the angle vertical to
one of the alternate interior Angles is a
corresponding angle to the other alternate
interior angle.
L1 and L2 are coplanar, with transversal T. ?A is
vertical to ?B, and ?B is an alternate interior
angle to ?C. therefore, ?A is a Corresponding
angle to ?C.
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Theorem 1
The AIP theorem
This theorem says that if two alternate interior
angles are congruent, then the lines that make
them are parallel. This is used to Prove two
lines parallel in a proof. This can be proved
because if ?A is congruent to ?C, and ?A is
supplementary to ?B because Of the Linear Pair
Theorem, so ?B is supplementary to ?C. Because
of theorem 9-8, which states that if a pair of
same-side Interior angles are supplementary,
then the lines are parallel, L1 and L2 are
parallel.
In simpler terms, if ?A and ?C are congruent,
then L1 and L2 are parallel.
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Given ?A is congruent to ?B
Proof of AIP theorem
Prove L1?L2
L1
A
B
L2
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The CAP theorem
The CAP theorem The CAP theorem, short for the
Corresponding Angle Parallel Theorem, States
that given two lines with a transversal through
them, if two corresponding angles are Congruent,
then the two lines are parallel.
T
A
If ?A is congruent to ?B, Then L1 is parallel
to L2.
L1
B
L2
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Given ?A is congruent to ?B
Prove L1L2
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Triangles
Key Terms
Right Triangle A right triangle is a triangle
with one right angle (90?). Because of This, we
can conclude that the two other angles are acute,
because all of the angles In a triangle must add
up to 180 degrees. Thus, No other angle can be
90? or higher, Because that would exceed this
rule of triangles. There are Many unique
properties About a right triangle and its
sides/angle measurements.
Acute angles
90?
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Hypotenuse The hypotenuse is the side opposite
of the right angle in a right triangle. It is
always longer than the two other sides of the
triangle. The ancient mathematician Pythagoras
found out that if the lengths of the two other
sides of the right Triangle were each squared
and then added together, the answer would be the
length of the hypotenuse squared.
Pythagorean Theorem A²B²C²
Hypotenuse
C
A
B
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Triangles
The angles of a triangle theorem
This theorem states that all of the angles of a
triangle add up to 180. There is no Way to prove
this theorem, but it is possible to prove that
all of the angles of a Triangle measure up to
less than 181?.
Given ?A and ?B are complementary
Prove ?C is right
1 ?A is comp. To ?B 2 m?a m?b90? 3
m?am?bm?c180? 4 ??c90?180? 5 ?c90? 6 ??C
is right
1 Given 2 Defn. of comp. 3 Angles of a
triangle thm. 4 Substitution 5 Subtracti0n
prop. Of 6 Defn. of right angle
Not real name
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Acute angles of a Right triangle theorem
This theorem states the acute angles of a right
triangle are complimentary. This is because the
angle of a triangle add up to 180?. Since one of
the angles is 90 Degrees, thats 90 off the 180
requirement. Thus, the other angles must add up
to be 90 degrees, because the sum of the angles
of any given triangle must add up to be 180
degrees. Because they add up to 90, the other
angles are complimentary.
X90-a, and a90-x.
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A
C
B
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Quadrilaterals
Key Terms
Quadrilateral ok, draw 4 coplanar points (lets
use p, q, r, and s), no three of them being
collinear. Then Connect them in consecutive order
(segments pq, qr, rs, and ps). Viola! Your very
own quadrilateral. Your mom will be proud.
Base sides
medians
Trapezoid a trapezoid is a quadrilateral with
one and only one pair of sides That are parallel.
The parallel sides are called the base sides, and
the Nonparallel sides are called the medians.
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Parallelogram a parallelogram is a quadrilateral
with all of the opposite sides Being parallel.
Thus, all the sides are equal. Also, there are
several other Properties in a parallelogram
because the opposite sides are parallel.
Rectangle a rectangle is a parallelogram with
right angles. Nothing more.
Right angles
Rhombus a rhombus take a parallelogram. Give it
equal sides. Viola! A Rhombus.
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Square a square is a combination of a rectangle
and a rhombus. It has Congruent, parallel sides,
the diagonals bisect each other and
are Congruent, and the angles are right.
Big Square
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The Opposite sides of a Parallelogram Theorem
this theorem says that the Opposite sides of a
parallelogram, which are parallel, are of equal
length. This is true because the diagonals of a
parallelogram divide the parallelogram Into two
congruent triangles, and since the corresponding
parts of the Triangles are congruent, the sides
are congruent.
Because this thing is a parallelogram, the
opposite sides are congruent.
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E
Given ABCD is a parallelogram, ?E??F, ?ADE? ?CBF
A
D
Prove ADE?BCF
S
B
R
C
1 Givens 2 ADBC 3 ?ADE?BCF
1 Given 2 Opposite sides of a gram 3 SAA
Postulate
F
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The Opposite angles of a gram Theorem
This theorem is similar to the last one the
opposite angles of a Parallelogram are equal in
measure. This is easy to prove because of The
diagonals of a parallelogram theorem the
diagonals of a gram Divide it into 2 congruent
triangles. Then you can just take the
ensuingcongruent triangles and compare the
corresponding angles, which are Congruent.
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A
D
Given ABCD is a parallelogram
Prove Theorem 9-14
S
R
B
C
1 ABCD is a gram 2 AD ? BC, AB ? CD 3 ?A ?
?C 4 ?ABD ? BCD
1Given 2 Opp. Sides of gram 3 Opp. ?s of
gram\ 4 SAS Postulate