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Chapter 9 Summary
Project
Parallelism
Triangles
Quadrilaterals
By Ernest Lee
12/17/03
Geometry Honors
Period 2
skew lines-lines that do not lie in the
same plane (AB, DE)
parallel lines-lines that are coplanar and
never intersect (AB, CD)
transversal-a line that intersects 2
coplanar lines at 2 different points
G
B
F
A
E
C
D
alternate interior angles-given lines L1 and L2 cut by transversal T at
points P and Q, let A be a point on L1 and let B be a point on L2 so that
A and B are on opposite sides of T, APQ and PQB are alternate
interior angles
A
Q
T
P
L1
B
L2
corresponding angles-given 2 lines cut by a transversal, if x and y
are alternate interior angles, and y and z are vertical angles, then
x and z are corresponding angles
y
z
y
w
v x
x
interior angles on the same side of the transversal-given 2 lines cut
by a transversal, if x and y are alternate interior angles, v and
w are alternate interior angles, and v and x form a linear pair,
then x and w are interior angles on the same side of the
transversal
intercept-if a transversal T intersects 2 lines L1 and L2 at points A
and B, then L1 and L2 intercepts segment AB on the transversal
T
A
B
L1
L2
The AIP Theorem
Given 2 lines cut by a transversal. If a pair of alternate
interior angles are congruent, then the lines are parallel.
a
P
b
Q
L1
L2
T
In the diagram, given lines L1 and L2 cut by the transversal T.
If ab, then L1 L2.
The CAP Theorem
Given 2 lines cut by a transversal. If a pair of corresponding
angles are congruent, then the lines are parallel.
P
In the diagram, given lines L1
and L2 cut by the transversal T. If
xy, then L1 L2.
y
Q
T
x
L1
L2
The PAI Theorem
If 2 parallel lines are cut by a transversal, then the alternate
interior angles are congruent.
P
In the figure, given L1L2 and
they are cut by the transversal T.
Then 12 and 34.
1
3
Q
L1
4
2
L2
T
The PCA Corollary
If 2 parallel lines are cut by a transversal, each pair of
corresponding angles are congruent.
In the figure, given L1L2 and
they are cut by the transversal T.
Then 15, 26, 37, and
48.
1
4
5
8
T
2
3
L1
6
7
L2
The AIP Theorem
Given: AD bisects CAB and
CA=CD
Prove: CDAB
C
D
x
y
A
z
B
S
R
1. AD bisects CAB, CA=CD
1. Given
2. xy
2. ITT
3. yz
3. def. of bisector
4. xz
4. TPE
5. CDAB
5. AIP
The CAP Theorem
Given: AC=BC and DCEB
Prove: CEAB
D
C
A
E
B
S
R
1. AC=BC, DCEB
1. Given
2. AB
2. ITT
3. ADCE
3. TPE
4. CEAB
4. CAP
The PAI Theorem
Given: CDAB and E is the midpoint of CB
Prove: AB=CD C
D
E
A
B
S
R
1. CDAB, E is the midpoint
of CB
1. Given
2. EC=EB
2. def. of midpoint
3. CEDBEA
3. VAT
4. BC
5. ∆EAB∆EDC
6. AB=DC
4. PAI
5. ASA
6. CPCTC
The PCA Corollary
Given: RT=RS and PQRS
Prove: PQ=PT
T
P
R
Q
S
S
R
1. RT=RS, PQRS
1. Given
2. TS
2. ITT
3. PQTS
3. PCA
4. TPQT
4. TPE
5. PQ=PT
5. ITT Converse
concurrent-2 or more lines are concurrent if there is a single point
that is on all of them
point of concurrency-the point shared by the concurrent lines is
the point of concurrency
The 30-60-90 Triangle Theorem
If an acute angle of a right triangle is 30, then the side
opposite the angles is half as long as the hypotenuse.
B
60
In the figure, given
∆ABC with a right angle at C and
mA is 30, then BC=1/2AB.
A
30
C
The Median Concurrence Theorem
The medians of every triangle are concurrent. Their point
of concurrency is 2/3rds of the way along each median, from the
vertex to the opposite side.
In the figure, given ∆ABC,
let D be the midpoint of CB, E be the
midpoint of AB, and F be the
midpoint of AC. Then the medians
AD, BF, and CE intersect at P so that
AP=2/3AD, BP=2/3FB, and CP=2/3CE.
C
F
A
P
E
D
B
The 30-60-90 Triangle Theorem
In ∆KMN, M is a right angle and mk=30. RS, TV,
and XY are each perpendicular to KM. If KR=6, KT=10, KX=13,
and KN=16, what are RS, TV, XY, and MN? What theorem did
you use?
N
X
T
R
30
K
S
V
Y
M
Using the 30-60-90 Triangle Theorem, we get
RS=1/2KR, TV=1/2KT, XY=1/2KX, and MN=1/2KN. After
plugging in values given, you get RS=1/2(6), TV=1/2(10),
XY=1/2(13), and MN=1/2(16). Therefore, RS=3, TV=5, XY=6.5,
and MN=8.
The Median Concurrence Theorem
In the figure, AD and CE are medians. If AD=15 and
CE=12, what are AP and CP?
C
P
A
E
D
B
Using the Median Concurrence Theorem, we can find
that AP=2/3AD and CP=2/3CE. By plugging in the numbers, you
get AP= 2/3(15) and CP= 2/3(12). Therefore, AP=10 and CP=8.
quadrilateral-the union of 4 segments that intersect only at their
endpoints
sides-the 4 segments that make up the quadrilateral
vertices-the endpoints of the segments that make up the quadrilateral
convex-a quadrilateral is convex if no 2 of its vertices lie on
opposite sides of a line containing a side of the quadrilateral
opposite sides-sides of a quadrilateral that never intersect
opposite angles-angles that do not have a side of the quadrilateral in
common
consecutive sides-sides that have a common endpoint
consecutive angles-angles that have a side of the quadrilateral in
common
diagonal-a segment in a quadrilateral that join 2 nonconsecutive
vertices
parallelogram-a quadrilateral in which both pairs of opposite sides are
parallel
trapezoid-a quadrilateral in which only one pair of opposite sides are
parallel
bases-the parallel sides of the trapezoid
median-the segment joining the midpoints of the nonparallel sides
rhombus-a parallelogram all of which whose sides are congruent
rectangle-a parallelogram all of
whose angles are right angles
square-a rectangle all of whose
sides are congruent
Each diagonal separates a parallelogram into two
congruent triangles.
D
A
B
C
In the figure, given ABCD is a parallelogram, then
∆BAD∆DCB.
In a parallelogram, any two opposite sides are
congruent.
D
A
B
C
In the figure, given ABCD is a parallelogram, AD=BC
and AB=CD.
In a parallelogram, any two opposite angles are congruent.
D
A
B
C
In the figure, given ABCD is a parallelogram, AC
and BD.
Given a quadrilateral in which both pairs of opposite sides are
congruent, then the quadrilateral is a parallelogram.
D
A
B
C
In the figure, given AD=BC and AB=CD, then ABCD is a
parallelogram.
In a parallelogram, any two opposite
angles are congruent.
Given a parallelogram ABCD with mb=4x+15
and mD=6x-27. Find the measures of the four angles.
What theorem did you use?
D
C
6x-27
4x+15
A
B
Because we know that any two opposite angles are
congruent in a parallelogram, we know that mA=mC and
mB=mD. 6x-27=4x+15, 2x=42, and x=21, so mB=mD=99.
Because A is supplementary to D and B is supplementary
to C, mA=mC=81.
Opposite sides are congruent, diagonal divides parallelogram
into congruent triangles, if opposite sides are congruent, the
quadrilateral is a parallelogram
R
S
Given: PQRS is a parallelogram and
PW=PS and RU=RQ
Prove: SWQU is a parallelogram.
U
W
P
S
R
1.PQRS is a parallelogram,
PW=PS, RU=RQ
1. Given
2. PS=RQ,
SR=PQ
3. PW=RU
2.
4. ∆SPR∆QRP
4.
5. SPWQRU,QPWSRU
6. ∆SPW∆QRU
7. SW=QU
Q
Opposite sides of a parallelogram are
congruent
3. TPE
Diagonal separates parallelogram into 2  ∆’s
5. CPCTC
6. SAS
7. CPCTC
8. ∆PQW∆RSU
9. QW=SU
8. SAS
10. SWQU is a parallelogram
10.
9. CPCTC
If both pairs of opposite sides are , then the
quadrilateral Is a parallelogram
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